Understanding Tangent Planes and Normals

So we finally get to it. We finally get to the idea of how we make a tangent that's a tangent figure, a tangent that's not just in a direction. not just along the x, not just along the y, not just along a vector that we call unit vector u. We think about a calc one idea really truly in three dimensions. We take this idea and we say, well, what do we want to do with surfaces? We want to find a tangent, but not just a line and a direction. A tangent, oh, if it's a tangent on a surface, it's got to be a tangent plane. What we're going to do right now, we know they exist. We've talked about them. The goal right now is to figure out how to find a tangent plane to a surface at a point. In Calc 1, the idea is take a curve, find the tangent line, tangent line to a curve at a point. Now, it's take a surface. Let's find a tangent plane. Let's see. to the surface at a point. So this tangent plane touches the surface at exactly one point within a neighborhood. That's what we are talking about here. Show advantage in the concept. So calc one, tangent line. Calc three, tangent plane. Now, please don't misconstrue this. We have found equations of tangent lines on surfaces, but they always had to be restricted, like along a certain direction, along a certain path, along a certain vector. Now we're saying, in order for us to find a tangent plane, look. Look, tangent planes have an unlimited amount of vectors in them. Did you guys get that? Infinite number. However, if we're going to find a tangent plane. What one thing besides a point do we need to define a plane? Normal vector. Our whole class is going to be spent on how we find that normal vector. Because if I can find a normal vector that defines that tangent plane, I can define a tangent plane. Do you understand the idea? Goal before, tangent line. Goal now, tangent plane. What do we need for planes? Normal vectors. Let's find the normal vector. You with me? I'm going to prove it. I'm going to go, it's going to take a little while. I know that. Some of you are like, ah, I hate proving things. Get used to proofs if you're going to be a math, science, engineering sort of major because understanding the proof is really good for understanding how to do the material. So we're going to get through it. So let's consider a surface, this surface. f of x, y equals z. So just some random surface. What dimension is this all in, everybody? I need more participation on this particular lesson. Say what? 3D. This is 3D. Okay. Or R3. Now here's the whole goal. This is going to build everything for us. If this is a surface in 3D, if I intersect it with a, let's call it horizontal, a plane that's parallel to the XY plane. If I intersect it with a plane, so say z equals 4, z equals 7, z equals any constant, that plane is going to cut into that surface. Does that make sense? It's going to create a what? Curve. Curve on that plane, on that level, it's called a, we've done it before. It's called level curve. So, if that's a 3D surface, if I restrict z to be a certain number, that certain number creates a horizontal plane over the xy plane, it's going to intersect the surface and create a level curve. Hadn't I let you understand the concept? That's what's going to happen. So this thing will have a level curve at the function equals a constant value. So here it is. Here's the service. I'm going to let this plane level curve, so I'm going to let this plane, this constant plane, intersect my surface. The plane's going to cut into the surface. It's going to create this curve that's on a level. I don't know what it looks like, but it's going to do this. It's going to create this level curve on the surface. Does that make sense? We are going to project that curve onto the xy plane. That's giving us a level curve. That's what we get out of this thing. So here's my surface. Let's just make up a... random curve. Here's my curve. Projected onto the xy plane. Now, there's a couple things I really want you to put in your head right now so the rest of the stuff makes sense. Are you following the surface I did? Surfaces are what dimension? What dimension are... If I take a plane and cut it and get a curve on the plane, then look on top and project it down onto the xy plane. So you get... So if I look from the top, project this, slice it, look down, project that plane. That's called a level curve. Project it onto the xy and get something that looks, I don't know what it looks like, but it looks like that. It looks like a curve. You with me? What dimension is it? 2D. 2D. Do you understand that level curves are two-dimensional? We talked about it before. You slice it, it's on a plane. It's 2D. It's 2D. Here's what we're going to do. So my idea is my level curve is always... I said some words last time and I meant them. I said some words that you need to understand about the last section. This level curve will always be one dimension less than the surface. If this was 4D, my level surface would be 3D. Does that make sense? It's always one dimension less. Let's do this. Let's represent this curve by a vector function. Can I do it? Yeah, of course I can. If it's continuous, this is curved, I can have this vector function. Oh my gosh, you thought they would never return. You're wrong. They do. They eat everyone. Vector function. Well, here's how vector functions look. They have an X component based on T. And the y component based on t. Verify something for me right now. That vector function is two-dimensional, yes or no? Because it's representing this thing on just the xy plane. So, long story made short, we had surface, we cut it with a plane. That's called a level curve. We project it onto an xy plane that's two-dimensional, one dimension less, and we're going to represent it by a vector function. Head on if you're with me. The exciting stuff's about to happen. Like, it's not exciting already, I know. But it's cool. So that represents the level curve of the surface. So the curve is represented by that one level curve. Now, add a little bit to it. That's a level curve at a certain value. So, it's at C. So, the function for the level curve, if we do this. It's a function with x's and y's equals c. Some of you guys are going to have some trouble seeing where that comes from. I want to show you where this comes from. Please watch. My function that I started with. Look at the board. Look right now because if you blink, you're going to miss me. This is my function, right? f of x, y. If I set z equal to a constant, c. If I set the z here, f of x, y equals c. This right here, it's not a surface. It is a level curve. Follow? I'm just saying, well, with that level curve, let's call that a vector function where we have this. Then x, the x component changes to x of t. That's what I'm doing. I'm substituting in the vector function into that function. And set it equal to c. Why? Because that right there creates for me this level curve. It's saying I'm holding the surface. Constant, C, to create a level curve defined by a vector function. Are you guys okay with that idea? That's what's happening. Just call it a vector function, then it changes my function. Basically, re-parameterizing my function, that one, in terms of a vector function because it's now a curve. That's the idea. So, if it's real, right, real. Now. That's still a level curve. You know that. It's not a surface because I'm holding this constant. It's a level curve. Let's take a derivative of both sides. If I take a derivative, if I take a derivative, Here's why we did this, okay? If I take a derivative, what's my independent variable now that I re-parameterized it by a vector function? So why would we do it? What's our derivative? Say it again. E with respect to t. What's the derivative of a constant? I don't even care what it is with respect to you. What's the derivative of a constant? Zero. Look, because we have a derivative of a function that has x's in it and y's in it and t's in it. Everyone focus right now so you get it. The main function is f. Did you get it? What is the only independent variable? What are x and y? Think back two sections. They're intermediate. I need the chain rule. I need to get from f to t. x and y are now intermediate variables. So, I cut it, I lay it down there, I call it a vector function. It lets me re-parameterize my function. So that's equal to a constant. Creating this curve, it's just 2d, it's just a curve. It lets me re-parameterize it so that I can take a derivative with respect to my 1, now 1. Independent variable. And it says, okay, how do we do it? You get from f to your independent variable. You go from your independent variable to your dependent variable. This should look very familiar because that right there is chain rule applied to two intermediate variables with one independent variable. Do you guys follow it? So, plus, okay, now I need to go from my main function to my... other intermediate variable I need to get from that guy to my independent. These are D's because there is only one independent. These are partials because there are two intermediate. I don't have time to go through that. It was in two sections ago, 13.5. Do you follow the idea? It's pretty cool, right? Do you guys get this from here to here, slicing it, making a curve, re-parameterizing it, and now you take a derivative and make a T? That's pretty freaking cool, right? But wait, there's more. This looks a lot like the result of... Oh my gosh, it has like a... You have two things multiplied, two things multiplied, and you're adding them together. That looks a lot like the result of maybe like a dot product. Let's think about this like a dot product again. If we do, if we go k, then maybe I would, oh yeah, equals zero, sorry. Maybe I would reverse these a little bit, but if I think about it this way, think about this as partial of f with respect to x, i. So, what do you think? plus partial of f with respect to yj dotted with dx, dt, i plus dy, dt, j. Now stop. I want you to make sure that you see this junk, okay? If you dot plot this, do you see that you get these two things multiplied together here? And these two things multiplied together here? And now if you see that. I have them a little out of order, but it's the same thing. But wait, you don't see it. I know you don't see it yet because you're, it doesn't look the same because we wrote different notation. But some of you guys, you have some magic happening right now. Okay? Watch. That's partial with respect to x. Partial is. That's beautiful! What is that? Come on, what is that? We just did the last section. We literally just finished it like 10 minutes ago. What is that? Gradient. That's the gradient. Dotted with, now wait, we're going to draw a lot of conclusions from this thing. What is that? Look at where this comes from. This is the derivative of x with respect to t plus the derivative of y with respect to t, and it's in a vector. It's in a vector. Look back up here, please. We redefine that as a vector function. Take the derivative of this. Do you get the, take the derivative of this whole thing actually. Do you get the derivative of x with respect to t? Yes. And i. Plus the derivative of y with respect to t. J. This is the derivative of that vector function. Furthermore, I know that those two things have a... Oh crap, here it comes! ...have a dot product that equals zero. This is amazing stuff, okay? We're just proving some... What? This is cool! So... So... I'm going to have to raise this side to get to the coolest... That's... so exciting! Here's your 30-second recap before we wrap this up with some very awesome statements and then end the class. They're going to take just a little bit of time, but they're worth it, okay? They're worth it when you have your mind on this stuff. Calc 1, tangent line. Calc 3, tangent plane. How do we do it? Take your surface, cut it with a level curve. A level curve is holding your surface constant at a plane. It's creating a literal curve that's a 2D projected on an xy plane, no problem. So I can look at it from the top. Got it. That can be represented by a vector function. How? Vector function with respect to t. Reparameterize as t, not a problem. It's one dimension less than the function you start with. Okay, well now that that's a vector function, reparameterize your main function you got in terms of t. That lets you take a derivative very quickly with the chain rule. This is just the chain rule. Constant, how we got our level curve goes to zero. Fantastic. That means that if we do this chain. rule, boom, boom, you can make a dot plug out of it. It's literally the same thing, but this is the gradient of that function that you start with. That's incredible. This is the vector function that we have right there. That's pretty cool. Most importantly are the following results. You believe that's true? Based on what I've told you, yeah. I've proved it up to this point, so hopefully you do. Tell me something, man, 11.3. Tell me something about a dot, the vectors, verify that's a vector. Verify that. Tell me something about the vectors if I dot product them and the result is equal to zero. They're orthogonal. They're orthogonal. They're perpendicular. They're orthogonal. I guess you're good at this. What this means. What all of this means. Number one, we know this. The derivative of a vector function gives me a tangent vector. You follow? Gives me a tangent vector. Now, wait though, a tangent vector to what? The surface or the level curve? Level curve. That's the level curve, right? So it's giving me the tangent vector to the level curve. That's what that's giving me. On a slope of, sorry. When a dot product equals zero, the two vectors that make that dot product up are orthogonal. So, so, what does the gradient give us? Let's look at it. Let's look at it. Here's the whole shebang, okay? Here's the whole thing. Let's call this a piece of... the level curve to a surface. You follow? Let's call this the point. What's r prime? Let's be, I can't speak anymore. Let's let this be defined by a vector function giving us that. What's r prime of t give us? Come on, put it together. This is a vector function. It's just in two different. I have this R of t. What's the derivative of R of t give us here? The tangent vector. It gives me the slope of the tangent vector. It could give me the line, right? So it's giving me the slope basically right there. That's what r prime of t gives me. Are you following me? This is important. This is cool. Now, wait a second, though. We know that that dot product equals 0. So I know automatically that at that point, if the dot product is 0, here's my tangent vector. But wait. Del f, the gradient of f, is orthogonal. What's orthogonal mean? Where does it have to go? Oh, that's not good. That's what that gives us. So what the gradient actually gives us here, it gives us the normal, hey, if we get a vector function, you find the derivative, the derivative gives us the tangent at any point on that curve, follow? Well, that means the gradient is the normal to any point on a level curve. Notice what's happening. You stick it in your head. You put it right now. I made it very clear a couple of sections ago, our last section, that the gradient works one dimension. below the function. Do you remember that? Always one dimension below. The gradient works one dimension below. For this surface that we are, the F surface, for the F surface we have, the gradient is given us a normal two-dimensional to the level curve, one dimension below. If the surface is in 3D, it's giving us the normal to something in 2D. But if the surface is in 4D, it gives us a normal to something in 3D. That is impressive, and that's what we're getting to in this stuff. So what the gradient does, what this gives us is the normal. To a level curve at a point. Well, wait a minute, wait a minute. Didn't the gradient give us the vector for the steepest climb up a surface or something? Didn't it give us that? Didn't it give us that? Same thing. It's the same. Watch. If I take like a contour map and I do something like this. Here's this. Let's go from here. That's my point. What's the... you know that this is a level curve higher than this one, higher than this one, correct? What's the fastest way to get from here to here? This way! No? Doesn't make sense. The fastest... way to get from here to my next level curve, which would be the quickest way up the hill, do you guys see it? Would be to draw your tangent, take something perpendicular to it, and go directly from right at the hill, to go directly away from the level curve. That is something that's orthogonal. That's what we're talking about. If that's a... Level curve, that's a level curve. The gradient tells me the fastest way up a hill? Yeah, because the gradient works on contour maps. It works on one dimension below. It works on finding something normal to a level curve, something that's going as fast as possible away from a level curve. That's as fast as possible. That's impressive! You should be like, what? Crazy! That's awesome! I don't know, I find this stuff, I get jacked up on this stuff. This is awesome. So, yeah, it is given the steepest climb, it's given the direction of the steepest climb up a hill. But the fastest way to get up a hill is the fastest way away from a level curve, and that is perpendicular to the tangent of that level curve. That's what's going on. Do I need to write this out for you? Do you want me to? Real quick? Just a little bit? Okay, just a little bit. Oh, don't sigh at me. Don't you do that. Come on, I'm here for you. I brought you candy. Thank you. That's it. The fastest path of the hill, the direction of steepest climb, the gradient, is along the path perpendicular to a level curve, and that's exactly what I have proven that the gradient gives us. The gradient gives us a normal to a level curve. That is what I need you to know you can think about it this way the fastest way to get From one level curve to another on a contour plot. It's going to be perpendicular You have you can't you can't go around so I can get there as fast as possible to be perpendicular to the level curve curve, that's the gradient. I want you to notice something else before I let you go. Can you notice, I'm not going to do it, but can you notice how this proof could easily be extrapolated for more than two variables? Can you see that? That's going to be the idea. So the two takeaways we have from this whole shebang, I promise, are the last thing to make it right. Gradient of f of xy is the normal to a level curve. When I set my function equal to a constant, that creates a level curve. Gradient gives us a normal level curve. If we have this, this is the big deal. Extrapolate. If we have that, then the gradient, if we have a function of three independent variables, then the gradient is the normal. To a level what? Surface. Think about a level surface. A plane. You can't do this on a curve, but on something 3D, you can put a plane on it. Planes need normals. We need a normal for that plane. So we're going to get this to give us a normal to a level surface. When we set... A 4D function equal to a constant. It gives a level surface. We find a normal to the level surface. That normal is the normal to the tangent plane. That is the whole point. So now time for examples. Look, we had kind of an awesome time last time developing the method for how we're going to find tangent planes and normal lines. We're going to do an example here in just a little bit, finding the tangent line and normal line to this. I'll explain that in just a second. This is our starting spot. But what we determined, the all of last lesson was this. How do we find the normal to anything? What we do is we find the gradient. What the gradient, but it's weird though. The gradient, what it does, it gives a normal vector. Verify that the gradient is a vector. Yes, no. Yes. It gives a normal vector to a level curve or a level surface of the function. So it's always one dimension lower than the function you're dealing with. Does that make sense to you? So if I have something R3, it's going to give normals to curves in R2. If I have something R4, it's going to give normals to surfaces in R3. That's the whole idea here. So the gradient does give normals, but it's one dimension below. Whatever function you have. That was the idea. So our goal here is we've got to find tangent planes. We've got to find normal lines. Now everyone in class, right now, what do you have to have to make the equation of a plane? What do you have to have? Not just any vector. What type of vector? You've got to have a normal vector. What happens about lines? Where do you need to find a line? Point. A point and a... you've got to have a vector. To find a normal line, we need a normal vector. So verify this for me. Tangent planes need a normal vector. Normal lines need a vector that is normal, a normal vector. Both of these things need a normal vector. In fact, it's the same normal vector. If you think about that, if I got a surface and something that's tangent, I need a normal vector, right? I say, hey, find a normal line. It's the same vector. So both of these guys need a normal vector. And what in the world gives us a normal vector all the time? It's on the board. What gives us a normal vector? The gradient. So the gradient is this magic normal vector that we get to use for both the tangent plane, because that needs normal, and the normal line. Head nod if you're okay with that idea. So the normal that we need... This pen is really weak. I'll change this. I don't know if that's any better. Nope, not any better. It's just black today. So that's the idea. Now, a quick example on how this actually works. I'm going to go a little slowly through this one. After we do, everything's going to move a little bit quicker. So let's find a tangent line and a normal line to... This now we are building towards this notice something notice something. I'm not asking for the tangent plane right now Why not well we'll think about this? What dimension is that in right now? How many variables do you see? This is 2d right now. This is not a surface. It's literally just hyperboloids along the x-axis That's all I'm doing so this is kind of a buildup problem right now, so let's make a little notation this right here This is in 2d That guy's 2D. Understood? Hello, yes, no. Yes. Now, here's the issue. Please watch carefully. This is the whole technique. This is going to make sure that you get it. Here's the issue. If I find the gradient of this, well, firstly, it's not even in a function notation. How am I supposed to do that? I've got to have this equal to the function. But if I find the gradient of this, the gradient will be in what dimension if I could even do it? What would it be in? One. So here's the plan. Listen carefully to the plan. It's what we do for the rest of this section. Are you with me? Focus. Focus. Here's the plan. We're going to cheat. We're going to cheat. What we're going to do is we're going to create some function that's a dimension higher than this one. Follow. Track with me. We're going to create a function that's a dimension higher than that one. That would be in what dimension? Three. So R3. Then when I find a gradient of that, it'll move it back to? And if we find a function higher than this, and we treat this as a level. of that function, verify. Gradient is normal to level curve, right? So the whole idea is create a function that's one dimension higher, that this is a level curve to it. Do you follow? Then when I find a gradient of that big function, it gives me level, it gives me normal to level curves. It gives me normal to that. I find the normal to the curve that I want. So we cheat. We go one dimension higher, find a gradient, moves it back down. That's the whole plan. Show of hands if you understand the whole plan. That's it. That's it. That's basically it. So this is R2, yeah, yeah, yeah. What we're going to do is we're going to cheat. So we're going to make this. How to cheat. First use your cell phones on test. How to cheat. We pretend, pretend that this, it's not really pretending, it's forcing. We're going to do this. That this... Is a level curve to some surface in R3. Or 3D, but however you want to think about it, I really don't care. So let's try it. Let's try it. Here's how we do it. Here's the whole shebang. This is it. This is it. If we take this, let's build a function that this is a level curve up. Recall, recall that any time I have a surface, please watch this, it's important. Any time I have a surface, verify that's a surface with two independent variables. equal to a constant, that gives me a level curve. Do you see it? Every time, that gives me a level curve, because it intersects the surface with the plane. That creates a curve in 2D. That's how we got this set in the first place. So here's the magic. Just set your your function equal to whatever your variable, get all your variables on one side, whatever your variable's saying. Ignore the constant. You don't need the constant. So if we do that, what was our function gonna be? Our function should be x squared minus y squared. Come on, come on, stick with me here. Is that a surface? Yes. Yes. Hello, yes, no. Yes. Yeah. Now wait. If we set this equal to 16, will it give me a level curve? Yes. Yes. That, so verify. This is a level curve of this thing. Namely, when the constant is 16. Hey, do this for me. Can you plug in the.53 into this? Can you do that for me here real quick? What's 5 squared? minus 9. What? What that says, if we plug that in, it says that that particular point, it's on a surface, yeah, but it's also on the level curve that I'm about to slice. So here's surface, point is on the surface, point is on the level curve. That's the whole point. We go one dimension higher. This thing is in 3D. That thing's in 3D. So let's do it. This is the big idea, so let's do like a 30 second recap. Do you guys think that's 2D? Do you understand that if I create this thing as a surface, that this is one particular level curve of that surface? Do you recall that? Namely, the one I said earlier, if the function is equal to a constant, we get a level curve. Namely, this is the level curve when the function is equal to 16. Verify the point is on the surface. It has to be. You can plug it in. Verify that point is also on the level curve. Yeah. So now imagine something. Imagine now 2D, 3D, surface. Imagine we took a gradient of this thing. What do gradients give us? Normals to? Come on, this is exciting. Do gradients give us normals to the function? No. The gradients give us normal to what? Is this a level curve? Of this. So when I take the gradient of this It gives me the normals to what I want. So I pretend, I just build it up. I go one dimension higher with a function that when I take a gradient of a function, I get back to the level curve I'm giving. I'm giving. Basically, I take my level curve and I want to build it up so the gradient gives me back down to that. It's like cheating. That's how you cheat. That's pretty cool. So the big note. The guy you're giving is a level curve to this. This is a level curve to my function, which means that the gradient will give me the normals to that level curve. Thank you. Also, this is one less thing, then we get to do our example and we're all good. Since that gradient gives normals to level curves, this particular gradient gives us a normal to this particular level curve of that surface. It also says one more thing. Since that point is on our level curve, if I find the gradient at that point, it's giving me the specific normal to that specific level curve at that specific point. Does that make sense? I'll say that one more time because we had a little bit of a kind of fuzz in our head right there, a little bit of a distraction. So here's what this says. Verify this is on that curve for sure. And it's also on the surface. So if I find the gradient of this surface, it's giving me a normal to this level curve if I plug that point in. It's at that particular point at that particular level curve. That's the idea. So. Let's try it. Can you all right now find me the gradient of the surface we just created, please? It should be really easy. It should be really, really, really fast. Find the gradient. You should know what a gradient is at this point. How do you find a gradient? Partials. So we take the partial of f with respect to... What is that? 2. The x component. What is the x component? 2x. 2x. Beautiful. Gradient is a vector. Then what? Come on, come on, quickly. Minus 2y. Minus, okay, 2yj. Look at the board. Look at it right now. This was 2d, right? Level curve. This was 2d. We built it up. What's the gradient going to give us? Something in 3D or something in 2D? 2D. 2D. It's giving us back this regarding this thing. It's the normals to this thing. That's what it does. This puts it back to 2D. That's the whole point. Take something in the dimension that you're given. Build up a function one dimension higher. That way the gradient gives you back in the same. universe as a thing you're given and it's the normal to that particular curve. So here's what this says. Write this down. You need to know it. This is the, verify it's the vector. It's a vector. This is the normal vector. Listen carefully, please. I don't want you to miss this. This is the normal vector to any level curve of that surface. The C has no part in this. Was the C anywhere in this, was the 16 anywhere in this at all? So this is a family of normals. It's the normal vector to any level curve of this at any point. It's the point, the point that restricts the level curve in this case. It's the point that gives us a specific normal vector to a specific level curve. So if you understand that concept. Okay, I'm going to write out just a little bit and then we're going to go. So this is the vector to any level curve. These are the level curves of any, any level curve of this is going to be of this format. It's going to be the function equal to a constant. This right here gives me the normal vector to any of those, any of them. It's a family of normal vectors to all. of those level curves. As soon as you put the point in there, that gives us a specific level curve, namely the one at 16. It gives us a specific normal, namely the one of this level curve at that particular point. So let's go ahead and let's try that. As I'm writing this, maybe you can be writing it or plugging in the.53 right now. Hey What value would you plug in for the x? What would you do? 5. And for the y? 13. So we're going to get 10i minus 6j. Great, great. No way. What's that even mean? Let's go through it just for 20 seconds. Make sure you fully grasp the concept. I really want you to be honest about this. Let me know that you really understand it or not, okay? So the idea, one last time through before we just start crushing some examples. Do you understand that the gradient is always one dimension below any function? Do you get that? Yeah. So gradients give normals to one dimension below, either curves or surfaces depending on what you're given. So our idea here is when you're given whatever... whatever you're given, a curve or a surface. We're going to create a function that is one dimension higher. How we do it, we just treat all our variables as independent variables. It creates, well think about that. If this gives us something 2D, creating a function with those as independent variables gives us something in 3D. It always builds it up one dimension higher. Do you follow? Create something one dimension higher. Take a gradient, it will give us normal vectors to the level curves of that function. That's. in the dimension that we're given already. So we build it up, we take it down. That's what we're doing here. So the gradient does that. The gradient creates normals to these things, which must be level curves of surfaces of the function that we create. That's the plan. Then we go, okay, well, where's the C? It doesn't matter. It's in this point. This point is on that level curve. It's on the surface, but it's also on that particular level curve. So when we find the gradient, yeah, you know what? That's a family of normal vectors. to the level curves. That's what it is. So that's a normal vector to any level curve of that function we created. But that point, when I plug it in, that point gives me a specific level curve. So I plug it in and go, okay, if this is a normal vector to any level curve, then when I plug in the point, this is a specific vector, a normal vector to it, that specific level curve. That's what we're doing. So that's the normal vector to that. that curve at that point. That's the idea. Honestly, show of hands if you understand the concept there. So, what now? What now? Well, if we're trying to figure out tangent lines and normal lines, that's a normal vector, yeah? Let's go ahead and remember we're in 2D right now. We're in a 2D universe. Can you find the slope? For the normal, this is pretty easy. We're going to kind of fly through this. If this is in 2D, slope is rise over run, or slope is y over x. Y over x, that's our slope. Be very careful. Is this the slope for the normal or the tangent? Normal. This is the slope of the normal line. So if I want to find the normal line, y minus y1, that's 3, equals mx minus x1. And you solve it down, and you have the normal line to that curve at that point. What about the tangent? What about the slope of the tangent? What do you know about the slope of something that's normal compared to the slope of something that is tangent? What do you know about those two things? If we're just talking 2D, we're just talking about lines right now. What do you know about the normal slope and a tangent slope? What do you know about them? They're perpendicular. What about the slopes? Do you know how to find a perpendicular slope? I reciprocate it, and I... you flip it, and you flip it good. Flip it means you reciprocate it, flip it good... Flip means to change the sign, so flip it, flip it, good. So if I want to find the slope of the tangent line, it's going to be the reciprocal of this, positive 5 thirds. You're going to go, okay, well that's fantastic. It's still the same point, y minus 3 equals 5 thirds, x minus 5, but when we solve that we get a slightly different line. Actually, majorly different line. Thank you. Listen, this is just something to illustrate this concept really quickly about how we build a function up and then we take the function back down. Did I explain this well enough for you to understand it for real? Yes. To understand the necessity of creating a different function. function so that the gradient gives us back this curve as a level curve. Do you understand that concept? That way when the gradient gives us back this curve, this normal to a level curve, it's a normal to the curve that I've been given initially. I got to tell you, this doesn't happen very often. This is a very simplified case. Actually, tangent planes are a little bit easier than this. But this is weird because I gave you something in just 2D, and we can't find a tangent plane in 2D. Verify that for me. If you get something on a plane, you can't find a plane to that. It doesn't make sense. So we found a tangent line and a normal line to that. And it's not too bad if you're given the normal vector. So the big idea. You're going to be given a function. We're going to consider... We're going to write this problem so that the function I'm given is a level curve to a function one dimension higher. So we're going to raise the dimension, and then the gradient will give us the normal to that curve that I started with, back down in that dimension. I feel like I'm talking around this so much, but I want to make sure that you understand the concept. Are you sure that you... get it can we do a different example what's that It's a normal. That's what gradients do. A normal to the function or a normal to the level curves. Let's try some. Okay, write it down then focus up here so we can talk about it. Number one, think back to chapter 11, surfaces. Think back to that. Is that a surface? Yes. That's a surface in 3D. Verify that 3D. Three variables, 3D. No problem. You with me? What type of surface is that? or all the money in the world that I don't have. Two sheet. Oh, what now? Two sheet. How many negatives? Two. All squared. Two negatives, two sheet. Hyperboloid. Two sheet hyperboloid. That's what that thing is. We're about to find the normal vector to that two sheet hyperboloid at exactly that point. And now that you're with me. So, here's the plan. What we're going to do right now is we're going to build this up. We're going to go one dimension higher. So let's create a big function. I'm going to use that capital one to go one dimension higher. It's typically what we do. It doesn't super matter, but we're going to do that. Hey, are all of your variables focused on this? I'm going to try to teach you as I'm doing an example for the next couple that we're going to get. Are all my variables on one side? Yes. Get everything on one side, get the constant on the other side if you have one. Your function just has to be all of the variables. So let's create this bigger function as negative x squared plus y squared minus d squared. And now that you're with me on that one, that's my function. Now, let's check this out. You just said that's in 3D. What is this thing? What dimension is that in? If we're treating these as independent variables, that's a, I don't know what it is, but it's a something in 4D. Does that make sense? That's 3D, this is a, I don't know, in 4D. I don't know, but it's in 4D, or R4. In fact, I want you to do it just so it sinks in your head, do this. Can you please plug in the... Point 132132. Can you please do that just for your own edification? Plug in the point 132. Did you get a value? What value did you get? If you plug that point into that function, it's going to give you that number. Coincidence? No! No, it's not a coincidence! What it's saying to you is that, yeah, that point's on this weird 4D surface, alright? But it's also on the level surface. It's saying that when I plug that thing into here, it's giving me something on this level surface of this. Verify this. f of x, y, z... Equals 4. Is that a level surface? Yeah. Anytime you set the function equal to a constant, you're getting a level surface. Are you able to follow me on this stuff right now? Anytime you do that, you get a level surface. So if I plug this point into here and I get that number, it's saying, yeah, that point. It's on this function, but it's also on that level surface that I'm creating. We're kind of working backwards. It's doing the same thing. This says that's a level surface, yeah, for sure, and that point is on that level surface. So now, so that's a level surface, namely the one that contains that point. That's the whole point. So I want you to find the gradient of this function right here. Find the gradient right now. I'd like to make sure that you found that as your grading show hands-free day. That was like half of us. Left-siders, did you get this? Okay, the gradient should be pretty easy, right? Some partials are put, IJK, and we're practically done. Now, now, man, if you could answer what this next question, if you could answer this next question, I'm going to be so happy. I'm going to be so happy. Make me happy. Don't make me angry. What is that? Vector? It's a normal vector to? The level surface. Level surfaces. To a specific level surface? No! This is a normal vector to a family of level surfaces of this whatever in R4. What dimension is this in? Quickly, what dimension? This is the normal to the level surfaces in 3D. So this is the normal vector. To a family of level surfaces that needs all of them for the original, for the function that you created. Now let's see if you can put it together right, Cy. Come on, come on, stick with me. Is this, look, look at the board, is this one of the level surfaces to which this is a normal vector? Come on, stick with me. Is this one of the level surfaces to this thing of which this is a normal vector? Is it one of them? Namely, it's the one that I set this equal to 4, that that point is actually on. So this thing, yeah, it's a normal vector to all of them at any point. This is the normal vector. To every level curve of this function at any point, it's given us all of them. How do we find a specific one? Well, just plug it in. That's given to give you a normal vector for sure, because this is a normal vector to level surfaces. So when I plug in a point... Oops. And we get this out. Whatever this is going to be, this is going to give you the specific normal vector to that specific level curve. At that specific point. And that's the magic. That's the magic. Can you do it? Can you plug in those values? Have you plugged in those values already? Can you please read them off the side what you got? Minus 4k. Ready? Surface? Yep. Pretend it's a level curve. Yep, now that's a level curve of that, level surface of that. Gradient finds normals to level surfaces. Find the gradient. This is a normal to the level curves, all of them, every level curve, every level surface of the function that you just created. Surface? Yes. Function? So that that's a level curve? Check. Take a gradient, and now this gradient is the normal to all the level surfaces. Got it. Plug in your point. Now this is a normal to that specific level surface at that specific point. You just found the normal to a surface at a point. What do you need to make a tangent plane? Point. A point. Do you have a point? And a normal. Did you find a normal? Yes. You need a point in the gradient. That's it. That's the whole shebang. We did all of this stuff to get to this point right here. So if you want to find a tangent plane, the process is actually really easy. It's one more step after this. Do you... Do you have a point? Do you have a normal? A specific normal that just happens to be at that point. You've got a point, you've got a normal, you've got a tangent point. Also, you've got a point, you've got a normal, you've got a normal line. Lines need a vector and a point. Planes need a normal vector and a point. So tangent planes, normal lines, seriously just one more pretty easy step. A little bit of review, which would be probably good for your test if you don't remember how to make planes and lines. To make a plane, you need a normal and a point. And that plane looks like this. You put your vector stuff first, point stuff second, vector stuff first, point stuff second, components first, coordinates second. That's a plane. That's how you find any plane. Hey, hey, just make a scene again. What gives us this one? What gives us this? A gradient. What gives us normals? A gradient. What gives us points? It's going to be given to you. All you've got to worry about is finding the gradient and putting a point. Done. That's it. Then, of course, you're going to solve this. You may put it in, take that and put it in, like, standard form. Let's walk through this one last time. Lines need a point and a vector, yeah? Points can be given to you. What type of a vector does a normal line need? You're going to use the same vector twice. The same vector for a tangent, because you need a normal, but then the normal is also the vector for the normal line. Does that make sense to you? If you need a normal for a tangent plane and you want to put a line in the direction of that normal, that normal vector is the vector that we want. So for the normal line, you keep the same exact end. You even keep the same point. But we have just a little different way that we do it. That's one way that we can do it. That's the, what type of equation of a line right now? What is that? Parametric or symmetric? Symmetric. That's symmetric, that's right. Notice something. Please, please look. Do this first every time. Here's why. If I give you the plane, if you give me the plane, if you give me that equation as the plane, from right here, do you see that you can just put this over this line, this over this line, this over this line? It's so easy to get from a tangent plane to a normal line. It's super, super fast. If you prefer the parametric, of course, we can solve these, set them all equal to t, and solve it. And we have the parametric. That's it. I'll tell you what, we're going to do three examples. We are going to be, maybe four if we have time, then that's it. And we're actually going to go really, really fast. So I'm going to get through as many examples as I possibly can for you guys. Are you with me? Are you ready to follow me pretty quickly? Do your brains hurt? No more than usual? Oh good, alright, then we're good. So for the rest of our problems we are going to find tangent planes and normal lines for all of them. It's going to be super fun and a good time. You wouldn't rather be doing anything else the day before spring break, would you? I mean, come on. Do you want us to meet during spring break? Yeah, sure. We could have an Easter egg hunt, and inside we'll have little math problems. Yes. And candy. Okay, let's do it. Let's do it. Here's number one. Number one, number one, here we go. This is a surface in R3. Look at the board. That's a surface in R3. Let's create something in R4. Take all of your variables, put them on one side, and create a function equal to that variable side. Do you need the constants? No. Can you all tell me what function is this going to be equal to? That's it. That's it. Can you please notice if I plug this in, that's going to give me negative 4. Do you see it? Look at it. It gives you negative 4. It's saying this point is on that level surface of that function. I do want to point something out to you. Some people really love to... No, no, you have to do this. X, Y, Z minus, I'm sorry, plus 4 equals 0. And then your function X, Y, Z equals X, Y, Z plus 4. You go, that has to be. That has to be it. What that does, that puts the level curve or level surface kind of at the origin, but it doesn't matter, because if you still plug this in, you get negative four plus four equals zero. And if you take a gradient, you start doing derivatives, that four disappears. So if you want to put the constant in there, you can, but you do not have to, because this is still a level surface of that function, just like if I added the four or this would still be a level surface of that function. Does that make sense? Either way is good, it really doesn't matter. What matters is you understand that you've got to go one dimension higher by creating a different function. So that's number one. Number two, come on now, what are you gonna do? Why are we doing gradient? Because we're supposed to, because you said so, Leonard. Why are we doing the gradient? Find the normal. Because if this is my function, the gradient is going to give me normal. to what? The level surfaces in this case, and that's one of them. So let's go ahead, let's find the gradient. It's going to give you the normals to the level surfaces of that function. Please go ahead and find the gradient. So fans feel okay with that one? Told you we're going to move fast. Keep in your head though, do not do math for the sake of doing math. That's a horrible way to do math. Alright? This right here is the gradient. It is the normal vector to any level. Level surface of that function you just created. Does that make sense? This is one of the level surfaces to that function you created. Therefore, this is the normal to that. It's just at any point. What are we going to do now? Plug in the point. That way you find the specific normal. Are you guys kidding this? It's super cool. And it's super cool. It's kind of easy. And now if you're with me on it. Surface, yep, big function, yep. This is the normal to any level surface of this. This is the normal to a specific level surface, namely that one because our point was on that one. We plugged in that point. It's giving us normal to that surface. exact point that's exactly what we want to show fans if you're following on this stuff it's so cool it's awesome so now we're practically done what do you need to find a tangent plane A point. Do we have a point? And a normal vector. Do we have a normal vector? Yes. Done. That's it. What do you need to find a normal line? A point and a vector that happens to be normal. That's it. You can even do stuff like this. Because we're not dealing with length or distance or anything, we can simplify. So I'm going to call this normal vector. Let's divide by 2. I could do, or divide by negative 2 if you want. It doesn't, it's either way. Let's choose that. If we divide by 2 or multiply by 1 half, you can simplify the normal. So we got a normal. We got a point. If we have a normal and a point, we have a plane. If we have a normal and a point to a level surface, we have a tangent plane. That's fantastic. I did a major mistake there. What should go first, 2 or negative 1? It's always vector, I gotta make sure we're okay with this. I don't want to stall us or anything, but do you see where all this stuff came from? For sure. Okay. From here, I want to even simplify right now. I'm going to simplify eventually. I'd go straight to my line. Because the normal line has a normal vector and the same point, we have x minus 2 over negative 1 equals y plus 1 over positive 2 equals z minus 2 over negative 1. This over this, this over this, this over this, done. It's that quick. It's that easy. Can you follow what I'm talking about? I would prefer symmetric if I were you because you're done. You should leave it right there. It's easy. Could you take this and make it parametric if you wanted to or if you had to? Yes. Now, I do want you to simplify your planes. Please don't leave them. Either one of those will work. Typically, you're going to see this one with a positive number and not with a negative number. It's the same plane, same plane. But typically, we get the one with the positive, which means that these can switch around a little bit, but it's still the same line. Look, we've gone quickly through this, but I want to make sure that this is, man, at least well explained for you guys. Has it been? Yes. What is this plane? It's a tangent plane. We just managed to find a tangent plane of what? That. That. Which was a level surface of this thing. We created it to find that. That's it. We also found the normal so that that one line that's going to be perfectly perpendicular or orthogonal to our plane and shoot right through that point, that's that line. That is what we're supposed to do. I'm going to have you try one on your own. I'm not going to give you long. These things are going to be very, very long. are very quick. So I'm going to try one with you, maybe do like one on our own, and then we're going to be done. So, finally, the tangent plane and normal line of this 3-D-R-3 surface at this point. As you're going through it, try your best to think through the process. I'm giving you something in 3D. I want you to create a function of just the variables, no constants, in 4D. Then the gradient gives me something back in 3D that's the normal to all the level surfaces, this being one of them, this being one of the level surfaces. Therefore, it's the normal to that curve. If I plug in the point, it's normal to that curve at that point. Try that. Go for it. Again, I'm going to give you maybe two things. Were you able to create your function? Yes. Do you need the plus six? No. Let me be very specific here. If you don't put the plus 6, it's like you subtracted this over and you have a level curve, level surface at the negative 6. Does that make sense? If you put the plus 6, it's like you have a level surface at equaling 0. Either way, it's going to work out just fine for your discipline of math. Thank you. How do you feel about taking partial derivatives right now? Good practice, huh? You're welcome. You're especially going to like the practice on that one right there. Trust me. For your test coming up in like 25 minutes. I love you Leonard! Best ever! I know. It's because I bring you candy. And knowledge. I'm going to pause you here for the sake of time, just so we can get through this here real quick. How many people were able to create the function and find the gradient? Show of hands if you were able to find that. That's fantastic. That's really good. Now, next question. Do you understand that this... This right here, this is the equation for every normal vector to every level surface of this crazy looking function and that level, all the family level surfaces includes this one. So this is the equation for any normal vector to that surface. And now that you understand that. If I plug in the point, this is the normal vector to that specific level surface function at that point. And now you're going to be understanding that one. What do you need for a tangent plane? Come on, say it, say it, put it in your head. You need what? A point. Do we have a point? Yes. And a normal vector. Do we have a normal vector? Yes. Gradients are normal vectors. That's the magic of this section. Do I have to, right-siders only, do I have to use this as my normal vector? No. What could I do instead? Let's do that to make it nicer. Could you? Yeah. It's not a problem. You can always use that one. It'll be just fine. But to save some effort, let's divide everything by three and get... just another normal vector. So big function, level surface, this is the equation gradient, is always a normal to a level whatever. So this is the equation for the normal vectors to this or any level surface. This is at that point because we plugged it in. This is a normal. We can just shrink this by dividing by 3, and we have our point which is that. Plains Tape Normal vector component and point coordinates and put them together. Let's check our work here. Did you get the same thing I got? You don't have these interchanged, right? We got our vector coming first. From here, what are you doing? By the way, what is this, plane or line? Because they look really similar. What is that? It's plane. From here, I'm going straight to my normal line. If I can. If I wanted to, I could solve each of these for x, y, and z by setting each of these equal to t and get parametric. But symmetric is the same exact thing. You can keep it that way. That's fine. And now if you understand, that's a normal line. So we've got this weird surface. Don't know what it looks like. It's a weird surface. At that point, this is the line that shoots through that point that's perfectly orthogonal to a tangent plane at that point. The tangent plane itself is right here. Did anybody solve that down and get it in a little bit nicer format? Can you say what that is for me, please? X plus three y minus four z. Yeah, equals what? How much? Negative seven? I don't know, I didn't do it. Oh wait. Negative 7. I get negative 11. Yeah. You get negative 11? Negative 11. I get that. Check it out. Yes. Should the normal line of the first term be x plus? You know, I think I did that twice. Thanks. You know how something that looks weird and you can't catch it? Like on your tests. This is the thing you look for when you check your work. Stuff like that. You're welcome for this one. You're welcome. Again, people who are taking tests in 20 minutes. after our break. Yeah, normal line, got it. Tangent plane, got it. How else would you see this written? Well, that could be negative, minus, plus, positive. That could be the other way you could write that plane. Same plane, though. Same exact thing. So this, here's a surface. This sits flat. This plane sits flat. Now listen, sometimes books try to trick you, problems try to trick you. If you're ever given something that is explicitly defined, this is implicitly defined, this is explicitly defined. Z is solved for... in terms of x and y. Be smart about this. We're not going to finish this problem, but we're certainly going to do this one. Check this out. Check this out. What I've told you is if that's 3D, which it is in R3, it's dependent equals. two independent. You with me? Get all your variables on one side. That's what we've been doing all the time. Get all your variables on one side, constant on the other, and that becomes your function. So you have two choices here. Watch carefully. We could choose to make this that or Now, either one. Do you guys see that they're exactly the same? Pick one. It doesn't even matter. It doesn't matter. So, if we have this equal to 4, this is now our level surface to some function higher, or this is now our level surface to some function higher. That's what we'd be doing here. The function itself... Would be this or if you want to on your own, you can even check it out. Check it out. If you plug in, well, I'm not giving you a point, but if you plug in a point, this would equal that number. If you plug in a point, this would equal that number. This is a level surface. to this 4D curve, this is a 3D level surface, to this 4D curve, 4D surface, something, whatever that is. You guys understand what I'm talking about? This is what you'd find the gradient of, one of those two things. Also, you can be smart about it. You know how to take the. derivatives with ln, making this ln x minus ln y minus z, verify that you can do that, would make your work a lot easier. That'd be pretty nice. Do stuff like that. Do stuff like that. That's so convenient to do that. Then you find gradient, and you're practically done. Gradient is the normal to the original thing you're given if you build it one dimension higher. Let's try it with this one. What's the first thing I'm doing on that problem? Everybody in class right now, what's the first thing I'm doing on that problem? Variables one side, left side, how would I do it? Would you subtract z or would you subtract tangent? It doesn't matter. What did I do? I think I subtracted z. Let's go ahead. I want you to subtract z from both sides. Do you understand the reason why? I need my variables on one side to create that bigger function. It has to have all of our variables in it. So we get our variables on one side. Next, I want you to create the function. Take five seconds, create the function for me. Come on, quickly, create the function. Right-siders, what's the function that you, sounds funny coming up, what's the function? All right, what's the function that we get out of this thing? That's it. Is this super hard for you or pretty easy for you? Pretty easy? So, get everything on one side, that's your function. Why? Why? Why is that your function? Well, because if this is in 4D, then this right here, nothing. This one, this one is a level surface in 3D. The level surface, namely, when I set this equal to zero. If I set a function equal to a constant, it gives me a level surface. If I set this equal to zero, it gives me that level surface. That's the whole point. This is a level surface to that new function that you just created. Cool, huh? Now, what gives us normals to levels? What gives us normals to level surfaces? Come on, everybody. What gives us normals to level surfaces? So when I create this function to which this is a level surface, I get the gradient. It's the normal to the level surfaces. Find the gradient. You know what? Now, we've actually done this before. I'm going to, just for the sake of time, give you what the gradient is. This is that one where you do chain rule. You have chain rule with respect to x and y, and you divide by 1 plus this thing squared, and we simplify. This is, you should have to trust me on it, or maybe if you're doing this again, pause it, or if you want to look at your notes later, try it on your own. What this gives us is this. We've actually done it before in this class. Plus this. Minus that. Partial with respect to x is this thing. Partial with respect to y is this thing. Partial with respect to z, this is gone. It's just that thing. Gradient takes the partials, puts i, j, and k. That right there is the, you just have to trust me or do it on your own and verify that that is correct. Do you guys know how to do partial derivatives? Yes. Okay, we don't have time to practice that right now. What is it though? Come on, this is the big deal right now. What is this thing? Normals. Normals. Two. Any level surface. This is one of them. But this is one of them only at that point. So we got to plug in that point to this normal vector equation to get a specific normal vector to a specific level curve at a specific point. Let's plug that in. It looks harder than it is. There's no Z to plug in, so you don't have to deal with it. So it's negative 1 half i here, negative 1 half j, and then this doesn't change, so minus k. You still with me? Brains hurt more? Yes. Don't worry. You'll be fine. You'll be all right. What is that? A specific normal vector. A specific, I love it, a specific normal vector to that specific level surface at that specific point. What do you need for a tangent plane? Point normal. Point and? Normal vector. We got it. Would you use that normal vector? Say it again. Multiply everything. Let's multiply everything by 2. You can do that. Scalar multiples, you're just picking a normal. As I'm writing down the last example we're going to do in this section, I want you to find the tangent plane and normal line. Thank you. Let's just make sure the numbers are in the right spot and I'll be pretty happy here. Did you all get the negative 1, 1, negative 2, minus 1, minus 1, minus pi before? Hopefully, you did not, you did. From here, go straight to your normal line and you're done. After that, you might simplify it. Did anybody simplify that? Can you tell me what you got, maybe? I just might have to swipe plus 2z. It's pi over 2. Okay. Equals how much? Negative. Negative pi over 2. Or change your signs and have a positive pi over 2. Either one. They're both the same. Same thing. Can you try one? Let's try one on your own. Take this as very good practice for partial derivatives, please. This is why we are doing it right now. Can you please take this as very good practice for partial derivatives right now? Set up the main function. Find your gradient. That could be done. Bless you. Thank you. About every 20 seconds I'm going to put a new step up on the board. I just want you to make sure you're checking your work. Stick with me. Did you get your major function correct? The one that's one dimension higher, did you get that correct? Do you need that 3 in there? Could you? Yeah. Subtract it, it's fine, it doesn't matter. It's the same thing. It's getting the level surface. Did you get your first partial derivative with respect to x? Correct. Did you? Yes. Did I? Yes. Okay. You know where the y came from? Yes. What rule is that? Chain rule. What's chain rule? Chain rule gives us x. Oh, last one, we can't miss that. Oh, maybe. How much is it? Last time I could explain it, I'm going to milk this thing for all it's worth. That's the surface you want. That's a function that's one dimension higher. This is the equation for the normal vectors to any level surface of this one. This is a level surface of this one. Therefore, this... is the equation for the normal vectors to that surface at any point. I want them at a specific point. That's why I plug it in. What we're about to get is the specific normal vector to that specific level surface of this function at that exact point. Tangents need normals. Normal lines need normals. That's why we're finding the gradient, which always gives us normals to a level surface or a level curve. Goodness, that was a lot to say. If I plug that in, I get 3i plus 3k because that's a zero and that's gone. Did you get it? That's the normal. Let's go ahead. Let's find an easier normal. What's an easier normal? Quickly, what's an easier normal? Scalar multiples are super nice. We know to find tangent planes. We take vector component, x coordinate, vector component, y coordinate, vector component. Z coordinate equals 0. This whole thing gives us X plus Z equals 1. That's tangent plane. That's what we wanted. Did you find X plus Z equals 1? It's beautiful. From here, we got normal line as well. Please do it from here. Please don't do it from here. Please don't do that. When we have this, we go, okay, well, how are we supposed to do it? This gives us x over 1. This right here, you can't divide by 0. This is a weird one. So what we do with this, we do this. We do z minus 1 over 1. That's true. But this is one of those ones where that y has no t in it. The t is 0, so it's a constant y. Equals 3. And that's how we would write that. It's a comma. It's not equals. So we have x equals z, no problem, and then we get that y equals 3. Did it make sense to you? Are you sure you're okay with this? Easy, medium, or hard compared to the stuff we've been doing? Pretty easy. You know what gradient does now, right? Yeah.

We want to find a tangent, but not just a line and a direction. A tangent, oh, if it's a tangent on a surface, it's got to be a tangent plane. What we're going to do right now, we know they exist. We've talked about them.

The goal right now is to figure out how to find a tangent plane to a surface at a point. In Calc 1, the idea is take a curve, find the tangent line, tangent line to a curve at a point. Now, it's take a surface. Let's find a tangent plane. Let's see.

to the surface at a point. So this tangent plane touches the surface at exactly one point within a neighborhood. That's what we are talking about here.

Show advantage in the concept. So calc one, tangent line. Calc three, tangent plane.

Now, please don't misconstrue this. We have found equations of tangent lines on surfaces, but they always had to be restricted, like along a certain direction, along a certain path, along a certain vector. Now we're saying, in order for us to find a tangent plane, look.

Look, tangent planes have an unlimited amount of vectors in them. Did you guys get that? Infinite number. However, if we're going to find a tangent plane.

What one thing besides a point do we need to define a plane? Normal vector. Our whole class is going to be spent on how we find that normal vector. Because if I can find a normal vector that defines that tangent plane, I can define a tangent plane. Do you understand the idea?

Goal before, tangent line. Goal now, tangent plane. What do we need for planes? Normal vectors.

Let's find the normal vector. You with me? I'm going to prove it. I'm going to go, it's going to take a little while. I know that.

Some of you are like, ah, I hate proving things. Get used to proofs if you're going to be a math, science, engineering sort of major because understanding the proof is really good for understanding how to do the material. So we're going to get through it.

So let's consider a surface, this surface. f of x, y equals z. So just some random surface.

What dimension is this all in, everybody? I need more participation on this particular lesson. Say what?

3D. This is 3D. Okay.

Or R3. Now here's the whole goal. This is going to build everything for us.

If this is a surface in 3D, if I intersect it with a, let's call it horizontal, a plane that's parallel to the XY plane. If I intersect it with a plane, so say z equals 4, z equals 7, z equals any constant, that plane is going to cut into that surface. Does that make sense? It's going to create a what? Curve.

Curve on that plane, on that level, it's called a, we've done it before. It's called level curve. So, if that's a 3D surface, if I restrict z to be a certain number, that certain number creates a horizontal plane over the xy plane, it's going to intersect the surface and create a level curve. Hadn't I let you understand the concept? That's what's going to happen.

So this thing will have a level curve at the function equals a constant value. So here it is. Here's the service. I'm going to let this plane level curve, so I'm going to let this plane, this constant plane, intersect my surface.

The plane's going to cut into the surface. It's going to create this curve that's on a level. I don't know what it looks like, but it's going to do this. It's going to create this level curve on the surface. Does that make sense?

We are going to project that curve onto the xy plane. That's giving us a level curve. That's what we get out of this thing. So here's my surface. Let's just make up a...

random curve. Here's my curve. Projected onto the xy plane. Now, there's a couple things I really want you to put in your head right now so the rest of the stuff makes sense.

Are you following the surface I did? Surfaces are what dimension? What dimension are...

If I take a plane and cut it and get a curve on the plane, then look on top and project it down onto the xy plane. So you get... So if I look from the top, project this, slice it, look down, project that plane. That's called a level curve. Project it onto the xy and get something that looks, I don't know what it looks like, but it looks like that.

It looks like a curve. You with me? What dimension is it?

2D. 2D. Do you understand that level curves are two-dimensional?

We talked about it before. You slice it, it's on a plane. It's 2D.

It's 2D. Here's what we're going to do. So my idea is my level curve is always... I said some words last time and I meant them. I said some words that you need to understand about the last section.

This level curve will always be one dimension less than the surface. If this was 4D, my level surface would be 3D. Does that make sense? It's always one dimension less.

Let's do this. Let's represent this curve by a vector function. Can I do it?

Yeah, of course I can. If it's continuous, this is curved, I can have this vector function. Oh my gosh, you thought they would never return.

You're wrong. They do. They eat everyone. Vector function.

Well, here's how vector functions look. They have an X component based on T. And the y component based on t. Verify something for me right now. That vector function is two-dimensional, yes or no?

Because it's representing this thing on just the xy plane. So, long story made short, we had surface, we cut it with a plane. That's called a level curve.

We project it onto an xy plane that's two-dimensional, one dimension less, and we're going to represent it by a vector function. Head on if you're with me. The exciting stuff's about to happen.

Like, it's not exciting already, I know. But it's cool. So that represents the level curve of the surface.

So the curve is represented by that one level curve. Now, add a little bit to it. That's a level curve at a certain value. So, it's at C.

So, the function for the level curve, if we do this. It's a function with x's and y's equals c. Some of you guys are going to have some trouble seeing where that comes from.

I want to show you where this comes from. Please watch. My function that I started with. Look at the board.

Look right now because if you blink, you're going to miss me. This is my function, right? f of x, y. If I set z equal to a constant, c.

If I set the z here, f of x, y equals c. This right here, it's not a surface. It is a level curve. Follow?

I'm just saying, well, with that level curve, let's call that a vector function where we have this. Then x, the x component changes to x of t. That's what I'm doing.

I'm substituting in the vector function into that function. And set it equal to c. Why?

Because that right there creates for me this level curve. It's saying I'm holding the surface. Constant, C, to create a level curve defined by a vector function.

Are you guys okay with that idea? That's what's happening. Just call it a vector function, then it changes my function.

Basically, re-parameterizing my function, that one, in terms of a vector function because it's now a curve. That's the idea. So, if it's real, right, real. Now.

That's still a level curve. You know that. It's not a surface because I'm holding this constant.

It's a level curve. Let's take a derivative of both sides. If I take a derivative, if I take a derivative, Here's why we did this, okay?

If I take a derivative, what's my independent variable now that I re-parameterized it by a vector function? So why would we do it? What's our derivative?

Say it again. E with respect to t. What's the derivative of a constant?

I don't even care what it is with respect to you. What's the derivative of a constant? Zero.

Look, because we have a derivative of a function that has x's in it and y's in it and t's in it. Everyone focus right now so you get it. The main function is f.

Did you get it? What is the only independent variable? What are x and y?

Think back two sections. They're intermediate. I need the chain rule.

I need to get from f to t. x and y are now intermediate variables. So, I cut it, I lay it down there, I call it a vector function. It lets me re-parameterize my function. So that's equal to a constant.

Creating this curve, it's just 2d, it's just a curve. It lets me re-parameterize it so that I can take a derivative with respect to my 1, now 1. Independent variable. And it says, okay, how do we do it? You get from f to your independent variable.

You go from your independent variable to your dependent variable. This should look very familiar because that right there is chain rule applied to two intermediate variables with one independent variable. Do you guys follow it?

So, plus, okay, now I need to go from my main function to my... other intermediate variable I need to get from that guy to my independent. These are D's because there is only one independent.

These are partials because there are two intermediate. I don't have time to go through that. It was in two sections ago, 13.5. Do you follow the idea?

It's pretty cool, right? Do you guys get this from here to here, slicing it, making a curve, re-parameterizing it, and now you take a derivative and make a T? That's pretty freaking cool, right? But wait, there's more.

This looks a lot like the result of... Oh my gosh, it has like a... You have two things multiplied, two things multiplied, and you're adding them together.

That looks a lot like the result of maybe like a dot product. Let's think about this like a dot product again. If we do, if we go k, then maybe I would, oh yeah, equals zero, sorry. Maybe I would reverse these a little bit, but if I think about it this way, think about this as partial of f with respect to x, i. So, what do you think?

plus partial of f with respect to yj dotted with dx, dt, i plus dy, dt, j. Now stop. I want you to make sure that you see this junk, okay? If you dot plot this, do you see that you get these two things multiplied together here?

And these two things multiplied together here? And now if you see that. I have them a little out of order, but it's the same thing.

But wait, you don't see it. I know you don't see it yet because you're, it doesn't look the same because we wrote different notation. But some of you guys, you have some magic happening right now.

Okay? Watch. That's partial with respect to x. Partial is.

That's beautiful! What is that? Come on, what is that? We just did the last section. We literally just finished it like 10 minutes ago.

What is that? Gradient. That's the gradient.

Dotted with, now wait, we're going to draw a lot of conclusions from this thing. What is that? Look at where this comes from.

This is the derivative of x with respect to t plus the derivative of y with respect to t, and it's in a vector. It's in a vector. Look back up here, please. We redefine that as a vector function. Take the derivative of this.

Do you get the, take the derivative of this whole thing actually. Do you get the derivative of x with respect to t? Yes. And i. Plus the derivative of y with respect to t.

J. This is the derivative of that vector function. Furthermore, I know that those two things have a... Oh crap, here it comes!

...have a dot product that equals zero. This is amazing stuff, okay? We're just proving some... What?

This is cool! So... So... I'm going to have to raise this side to get to the coolest...

That's... so exciting! Here's your 30-second recap before we wrap this up with some very awesome statements and then end the class.

They're going to take just a little bit of time, but they're worth it, okay? They're worth it when you have your mind on this stuff. Calc 1, tangent line.

Calc 3, tangent plane. How do we do it? Take your surface, cut it with a level curve.

A level curve is holding your surface constant at a plane. It's creating a literal curve that's a 2D projected on an xy plane, no problem. So I can look at it from the top.

Got it. That can be represented by a vector function. How?

Vector function with respect to t. Reparameterize as t, not a problem. It's one dimension less than the function you start with. Okay, well now that that's a vector function, reparameterize your main function you got in terms of t.

That lets you take a derivative very quickly with the chain rule. This is just the chain rule. Constant, how we got our level curve goes to zero. Fantastic.

That means that if we do this chain. rule, boom, boom, you can make a dot plug out of it. It's literally the same thing, but this is the gradient of that function that you start with. That's incredible. This is the vector function that we have right there.

That's pretty cool. Most importantly are the following results. You believe that's true?

Based on what I've told you, yeah. I've proved it up to this point, so hopefully you do. Tell me something, man, 11.3. Tell me something about a dot, the vectors, verify that's a vector. Verify that.

Tell me something about the vectors if I dot product them and the result is equal to zero. They're orthogonal. They're orthogonal. They're perpendicular. They're orthogonal.

I guess you're good at this. What this means. What all of this means.

Number one, we know this. The derivative of a vector function gives me a tangent vector. You follow?

Gives me a tangent vector. Now, wait though, a tangent vector to what? The surface or the level curve? Level curve.

That's the level curve, right? So it's giving me the tangent vector to the level curve. That's what that's giving me.

On a slope of, sorry. When a dot product equals zero, the two vectors that make that dot product up are orthogonal. So, so, what does the gradient give us? Let's look at it. Let's look at it.

Here's the whole shebang, okay? Here's the whole thing. Let's call this a piece of...

the level curve to a surface. You follow? Let's call this the point. What's r prime? Let's be, I can't speak anymore.

Let's let this be defined by a vector function giving us that. What's r prime of t give us? Come on, put it together.

This is a vector function. It's just in two different. I have this R of t.

What's the derivative of R of t give us here? The tangent vector. It gives me the slope of the tangent vector.

It could give me the line, right? So it's giving me the slope basically right there. That's what r prime of t gives me. Are you following me? This is important.

This is cool. Now, wait a second, though. We know that that dot product equals 0. So I know automatically that at that point, if the dot product is 0, here's my tangent vector. But wait. Del f, the gradient of f, is orthogonal.

What's orthogonal mean? Where does it have to go? Oh, that's not good.

That's what that gives us. So what the gradient actually gives us here, it gives us the normal, hey, if we get a vector function, you find the derivative, the derivative gives us the tangent at any point on that curve, follow? Well, that means the gradient is the normal to any point on a level curve.

Notice what's happening. You stick it in your head. You put it right now.

I made it very clear a couple of sections ago, our last section, that the gradient works one dimension. below the function. Do you remember that?

Always one dimension below. The gradient works one dimension below. For this surface that we are, the F surface, for the F surface we have, the gradient is given us a normal two-dimensional to the level curve, one dimension below.

If the surface is in 3D, it's giving us the normal to something in 2D. But if the surface is in 4D, it gives us a normal to something in 3D. That is impressive, and that's what we're getting to in this stuff. So what the gradient does, what this gives us is the normal. To a level curve at a point.

Well, wait a minute, wait a minute. Didn't the gradient give us the vector for the steepest climb up a surface or something? Didn't it give us that? Didn't it give us that?

Same thing. It's the same. Watch. If I take like a contour map and I do something like this. Here's this.

Let's go from here. That's my point. What's the... you know that this is a level curve higher than this one, higher than this one, correct?

What's the fastest way to get from here to here? This way! No?

Doesn't make sense. The fastest... way to get from here to my next level curve, which would be the quickest way up the hill, do you guys see it?

Would be to draw your tangent, take something perpendicular to it, and go directly from right at the hill, to go directly away from the level curve. That is something that's orthogonal. That's what we're talking about.

If that's a... Level curve, that's a level curve. The gradient tells me the fastest way up a hill?

Yeah, because the gradient works on contour maps. It works on one dimension below. It works on finding something normal to a level curve, something that's going as fast as possible away from a level curve. That's as fast as possible. That's impressive!

You should be like, what? Crazy! That's awesome! I don't know, I find this stuff, I get jacked up on this stuff. This is awesome.

So, yeah, it is given the steepest climb, it's given the direction of the steepest climb up a hill. But the fastest way to get up a hill is the fastest way away from a level curve, and that is perpendicular to the tangent of that level curve. That's what's going on. Do I need to write this out for you?

Do you want me to? Real quick? Just a little bit? Okay, just a little bit.

Oh, don't sigh at me. Don't you do that. Come on, I'm here for you.

I brought you candy. Thank you. That's it.

The fastest path of the hill, the direction of steepest climb, the gradient, is along the path perpendicular to a level curve, and that's exactly what I have proven that the gradient gives us. The gradient gives us a normal to a level curve. That is what I need you to know you can think about it this way the fastest way to get From one level curve to another on a contour plot.

It's going to be perpendicular You have you can't you can't go around so I can get there as fast as possible to be perpendicular to the level curve curve, that's the gradient. I want you to notice something else before I let you go. Can you notice, I'm not going to do it, but can you notice how this proof could easily be extrapolated for more than two variables?

Can you see that? That's going to be the idea. So the two takeaways we have from this whole shebang, I promise, are the last thing to make it right.

Gradient of f of xy is the normal to a level curve. When I set my function equal to a constant, that creates a level curve. Gradient gives us a normal level curve.

If we have this, this is the big deal. Extrapolate. If we have that, then the gradient, if we have a function of three independent variables, then the gradient is the normal. To a level what?

Surface. Think about a level surface. A plane. You can't do this on a curve, but on something 3D, you can put a plane on it. Planes need normals.

We need a normal for that plane. So we're going to get this to give us a normal to a level surface. When we set... A 4D function equal to a constant. It gives a level surface.

We find a normal to the level surface. That normal is the normal to the tangent plane. That is the whole point.

So now time for examples. Look, we had kind of an awesome time last time developing the method for how we're going to find tangent planes and normal lines. We're going to do an example here in just a little bit, finding the tangent line and normal line to this. I'll explain that in just a second.

This is our starting spot. But what we determined, the all of last lesson was this. How do we find the normal to anything? What we do is we find the gradient. What the gradient, but it's weird though.

The gradient, what it does, it gives a normal vector. Verify that the gradient is a vector. Yes, no.

Yes. It gives a normal vector to a level curve or a level surface of the function. So it's always one dimension lower than the function you're dealing with. Does that make sense to you?

So if I have something R3, it's going to give normals to curves in R2. If I have something R4, it's going to give normals to surfaces in R3. That's the whole idea here.

So the gradient does give normals, but it's one dimension below. Whatever function you have. That was the idea.

So our goal here is we've got to find tangent planes. We've got to find normal lines. Now everyone in class, right now, what do you have to have to make the equation of a plane?

What do you have to have? Not just any vector. What type of vector? You've got to have a normal vector. What happens about lines?

Where do you need to find a line? Point. A point and a...

you've got to have a vector. To find a normal line, we need a normal vector. So verify this for me.

Tangent planes need a normal vector. Normal lines need a vector that is normal, a normal vector. Both of these things need a normal vector.

In fact, it's the same normal vector. If you think about that, if I got a surface and something that's tangent, I need a normal vector, right? I say, hey, find a normal line.

It's the same vector. So both of these guys need a normal vector. And what in the world gives us a normal vector all the time? It's on the board.

What gives us a normal vector? The gradient. So the gradient is this magic normal vector that we get to use for both the tangent plane, because that needs normal, and the normal line.

Head nod if you're okay with that idea. So the normal that we need... This pen is really weak. I'll change this.

I don't know if that's any better. Nope, not any better. It's just black today. So that's the idea.

Now, a quick example on how this actually works. I'm going to go a little slowly through this one. After we do, everything's going to move a little bit quicker.

So let's find a tangent line and a normal line to... This now we are building towards this notice something notice something. I'm not asking for the tangent plane right now Why not well we'll think about this?

What dimension is that in right now? How many variables do you see? This is 2d right now. This is not a surface.

It's literally just hyperboloids along the x-axis That's all I'm doing so this is kind of a buildup problem right now, so let's make a little notation this right here This is in 2d That guy's 2D. Understood? Hello, yes, no.

Yes. Now, here's the issue. Please watch carefully.

This is the whole technique. This is going to make sure that you get it. Here's the issue. If I find the gradient of this, well, firstly, it's not even in a function notation. How am I supposed to do that?

I've got to have this equal to the function. But if I find the gradient of this, the gradient will be in what dimension if I could even do it? What would it be in? One. So here's the plan.

Listen carefully to the plan. It's what we do for the rest of this section. Are you with me? Focus.

Focus. Here's the plan. We're going to cheat.

We're going to cheat. What we're going to do is we're going to create some function that's a dimension higher than this one. Follow.

Track with me. We're going to create a function that's a dimension higher than that one. That would be in what dimension? Three.

So R3. Then when I find a gradient of that, it'll move it back to? And if we find a function higher than this, and we treat this as a level.

of that function, verify. Gradient is normal to level curve, right? So the whole idea is create a function that's one dimension higher, that this is a level curve to it.

Do you follow? Then when I find a gradient of that big function, it gives me level, it gives me normal to level curves. It gives me normal to that.

I find the normal to the curve that I want. So we cheat. We go one dimension higher, find a gradient, moves it back down.

That's the whole plan. Show of hands if you understand the whole plan. That's it.

That's it. That's basically it. So this is R2, yeah, yeah, yeah.

What we're going to do is we're going to cheat. So we're going to make this. How to cheat.

First use your cell phones on test. How to cheat. We pretend, pretend that this, it's not really pretending, it's forcing.

We're going to do this. That this... Is a level curve to some surface in R3.

Or 3D, but however you want to think about it, I really don't care. So let's try it. Let's try it. Here's how we do it. Here's the whole shebang.

This is it. This is it. If we take this, let's build a function that this is a level curve up.

Recall, recall that any time I have a surface, please watch this, it's important. Any time I have a surface, verify that's a surface with two independent variables. equal to a constant, that gives me a level curve.

Do you see it? Every time, that gives me a level curve, because it intersects the surface with the plane. That creates a curve in 2D.

That's how we got this set in the first place. So here's the magic. Just set your your function equal to whatever your variable, get all your variables on one side, whatever your variable's saying.

Ignore the constant. You don't need the constant. So if we do that, what was our function gonna be? Our function should be x squared minus y squared. Come on, come on, stick with me here.

Is that a surface? Yes. Yes.

Hello, yes, no. Yes. Yeah. Now wait.

If we set this equal to 16, will it give me a level curve? Yes. Yes. That, so verify.

This is a level curve of this thing. Namely, when the constant is 16. Hey, do this for me. Can you plug in the.53 into this? Can you do that for me here real quick?

What's 5 squared? minus 9. What? What that says, if we plug that in, it says that that particular point, it's on a surface, yeah, but it's also on the level curve that I'm about to slice. So here's surface, point is on the surface, point is on the level curve. That's the whole point.

We go one dimension higher. This thing is in 3D. That thing's in 3D. So let's do it. This is the big idea, so let's do like a 30 second recap.

Do you guys think that's 2D? Do you understand that if I create this thing as a surface, that this is one particular level curve of that surface? Do you recall that? Namely, the one I said earlier, if the function is equal to a constant, we get a level curve. Namely, this is the level curve when the function is equal to 16. Verify the point is on the surface.

It has to be. You can plug it in. Verify that point is also on the level curve. Yeah. So now imagine something.

Imagine now 2D, 3D, surface. Imagine we took a gradient of this thing. What do gradients give us? Normals to? Come on, this is exciting.

Do gradients give us normals to the function? No. The gradients give us normal to what? Is this a level curve? Of this.

So when I take the gradient of this It gives me the normals to what I want. So I pretend, I just build it up. I go one dimension higher with a function that when I take a gradient of a function, I get back to the level curve I'm giving.

I'm giving. Basically, I take my level curve and I want to build it up so the gradient gives me back down to that. It's like cheating.

That's how you cheat. That's pretty cool. So the big note.

The guy you're giving is a level curve to this. This is a level curve to my function, which means that the gradient will give me the normals to that level curve. Thank you.

Also, this is one less thing, then we get to do our example and we're all good. Since that gradient gives normals to level curves, this particular gradient gives us a normal to this particular level curve of that surface. It also says one more thing.

Since that point is on our level curve, if I find the gradient at that point, it's giving me the specific normal to that specific level curve at that specific point. Does that make sense? I'll say that one more time because we had a little bit of a kind of fuzz in our head right there, a little bit of a distraction. So here's what this says. Verify this is on that curve for sure.

And it's also on the surface. So if I find the gradient of this surface, it's giving me a normal to this level curve if I plug that point in. It's at that particular point at that particular level curve.

That's the idea. So. Let's try it.

Can you all right now find me the gradient of the surface we just created, please? It should be really easy. It should be really, really, really fast.

Find the gradient. You should know what a gradient is at this point. How do you find a gradient?

Partials. So we take the partial of f with respect to... What is that? 2. The x component. What is the x component?

2x. 2x. Beautiful. Gradient is a vector. Then what?

Come on, come on, quickly. Minus 2y. Minus, okay, 2yj. Look at the board.

Look at it right now. This was 2d, right? Level curve. This was 2d. We built it up.

What's the gradient going to give us? Something in 3D or something in 2D? 2D. 2D. It's giving us back this regarding this thing.

It's the normals to this thing. That's what it does. This puts it back to 2D.

That's the whole point. Take something in the dimension that you're given. Build up a function one dimension higher.

That way the gradient gives you back in the same. universe as a thing you're given and it's the normal to that particular curve. So here's what this says. Write this down. You need to know it.

This is the, verify it's the vector. It's a vector. This is the normal vector. Listen carefully, please.

I don't want you to miss this. This is the normal vector to any level curve of that surface. The C has no part in this.

Was the C anywhere in this, was the 16 anywhere in this at all? So this is a family of normals. It's the normal vector to any level curve of this at any point.

It's the point, the point that restricts the level curve in this case. It's the point that gives us a specific normal vector to a specific level curve. So if you understand that concept.

Okay, I'm going to write out just a little bit and then we're going to go. So this is the vector to any level curve. These are the level curves of any, any level curve of this is going to be of this format.

It's going to be the function equal to a constant. This right here gives me the normal vector to any of those, any of them. It's a family of normal vectors to all.

of those level curves. As soon as you put the point in there, that gives us a specific level curve, namely the one at 16. It gives us a specific normal, namely the one of this level curve at that particular point. So let's go ahead and let's try that. As I'm writing this, maybe you can be writing it or plugging in the.53 right now. Hey What value would you plug in for the x?

What would you do? 5. And for the y? 13. So we're going to get 10i minus 6j. Great, great. No way.

What's that even mean? Let's go through it just for 20 seconds. Make sure you fully grasp the concept.

I really want you to be honest about this. Let me know that you really understand it or not, okay? So the idea, one last time through before we just start crushing some examples. Do you understand that the gradient is always one dimension below any function? Do you get that?

Yeah. So gradients give normals to one dimension below, either curves or surfaces depending on what you're given. So our idea here is when you're given whatever... whatever you're given, a curve or a surface.

We're going to create a function that is one dimension higher. How we do it, we just treat all our variables as independent variables. It creates, well think about that. If this gives us something 2D, creating a function with those as independent variables gives us something in 3D. It always builds it up one dimension higher.

Do you follow? Create something one dimension higher. Take a gradient, it will give us normal vectors to the level curves of that function.

That's. in the dimension that we're given already. So we build it up, we take it down.

That's what we're doing here. So the gradient does that. The gradient creates normals to these things, which must be level curves of surfaces of the function that we create. That's the plan. Then we go, okay, well, where's the C?

It doesn't matter. It's in this point. This point is on that level curve. It's on the surface, but it's also on that particular level curve. So when we find the gradient, yeah, you know what?

That's a family of normal vectors. to the level curves. That's what it is. So that's a normal vector to any level curve of that function we created.

But that point, when I plug it in, that point gives me a specific level curve. So I plug it in and go, okay, if this is a normal vector to any level curve, then when I plug in the point, this is a specific vector, a normal vector to it, that specific level curve. That's what we're doing. So that's the normal vector to that.

that curve at that point. That's the idea. Honestly, show of hands if you understand the concept there. So, what now?

What now? Well, if we're trying to figure out tangent lines and normal lines, that's a normal vector, yeah? Let's go ahead and remember we're in 2D right now. We're in a 2D universe. Can you find the slope?

For the normal, this is pretty easy. We're going to kind of fly through this. If this is in 2D, slope is rise over run, or slope is y over x.

Y over x, that's our slope. Be very careful. Is this the slope for the normal or the tangent?

Normal. This is the slope of the normal line. So if I want to find the normal line, y minus y1, that's 3, equals mx minus x1. And you solve it down, and you have the normal line to that curve at that point.

What about the tangent? What about the slope of the tangent? What do you know about the slope of something that's normal compared to the slope of something that is tangent? What do you know about those two things? If we're just talking 2D, we're just talking about lines right now.

What do you know about the normal slope and a tangent slope? What do you know about them? They're perpendicular.

What about the slopes? Do you know how to find a perpendicular slope? I reciprocate it, and I...

you flip it, and you flip it good. Flip it means you reciprocate it, flip it good... Flip means to change the sign, so flip it, flip it, good.

So if I want to find the slope of the tangent line, it's going to be the reciprocal of this, positive 5 thirds. You're going to go, okay, well that's fantastic. It's still the same point, y minus 3 equals 5 thirds, x minus 5, but when we solve that we get a slightly different line.

Actually, majorly different line. Thank you. Listen, this is just something to illustrate this concept really quickly about how we build a function up and then we take the function back down.

Did I explain this well enough for you to understand it for real? Yes. To understand the necessity of creating a different function. function so that the gradient gives us back this curve as a level curve. Do you understand that concept?

That way when the gradient gives us back this curve, this normal to a level curve, it's a normal to the curve that I've been given initially. I got to tell you, this doesn't happen very often. This is a very simplified case. Actually, tangent planes are a little bit easier than this. But this is weird because I gave you something in just 2D, and we can't find a tangent plane in 2D.

Verify that for me. If you get something on a plane, you can't find a plane to that. It doesn't make sense.

So we found a tangent line and a normal line to that. And it's not too bad if you're given the normal vector. So the big idea.

You're going to be given a function. We're going to consider... We're going to write this problem so that the function I'm given is a level curve to a function one dimension higher. So we're going to raise the dimension, and then the gradient will give us the normal to that curve that I started with, back down in that dimension. I feel like I'm talking around this so much, but I want to make sure that you understand the concept.

Are you sure that you... get it can we do a different example what's that It's a normal. That's what gradients do.

A normal to the function or a normal to the level curves. Let's try some. Okay, write it down then focus up here so we can talk about it.

Number one, think back to chapter 11, surfaces. Think back to that. Is that a surface?

Yes. That's a surface in 3D. Verify that 3D. Three variables, 3D.

No problem. You with me? What type of surface is that? or all the money in the world that I don't have. Two sheet.

Oh, what now? Two sheet. How many negatives?

Two. All squared. Two negatives, two sheet.

Hyperboloid. Two sheet hyperboloid. That's what that thing is. We're about to find the normal vector to that two sheet hyperboloid at exactly that point.

And now that you're with me. So, here's the plan. What we're going to do right now is we're going to build this up. We're going to go one dimension higher. So let's create a big function.

I'm going to use that capital one to go one dimension higher. It's typically what we do. It doesn't super matter, but we're going to do that.

Hey, are all of your variables focused on this? I'm going to try to teach you as I'm doing an example for the next couple that we're going to get. Are all my variables on one side? Yes.

Get everything on one side, get the constant on the other side if you have one. Your function just has to be all of the variables. So let's create this bigger function as negative x squared plus y squared minus d squared. And now that you're with me on that one, that's my function. Now, let's check this out.

You just said that's in 3D. What is this thing? What dimension is that in? If we're treating these as independent variables, that's a, I don't know what it is, but it's a something in 4D.

Does that make sense? That's 3D, this is a, I don't know, in 4D. I don't know, but it's in 4D, or R4.

In fact, I want you to do it just so it sinks in your head, do this. Can you please plug in the... Point 132132. Can you please do that just for your own edification? Plug in the point 132. Did you get a value?

What value did you get? If you plug that point into that function, it's going to give you that number. Coincidence?

No! No, it's not a coincidence! What it's saying to you is that, yeah, that point's on this weird 4D surface, alright?

But it's also on the level surface. It's saying that when I plug that thing into here, it's giving me something on this level surface of this. Verify this. f of x, y, z... Equals 4. Is that a level surface?

Yeah. Anytime you set the function equal to a constant, you're getting a level surface. Are you able to follow me on this stuff right now?

Anytime you do that, you get a level surface. So if I plug this point into here and I get that number, it's saying, yeah, that point. It's on this function, but it's also on that level surface that I'm creating.

We're kind of working backwards. It's doing the same thing. This says that's a level surface, yeah, for sure, and that point is on that level surface.

So now, so that's a level surface, namely the one that contains that point. That's the whole point. So I want you to find the gradient of this function right here. Find the gradient right now. I'd like to make sure that you found that as your grading show hands-free day.

That was like half of us. Left-siders, did you get this? Okay, the gradient should be pretty easy, right?

Some partials are put, IJK, and we're practically done. Now, now, man, if you could answer what this next question, if you could answer this next question, I'm going to be so happy. I'm going to be so happy.

Make me happy. Don't make me angry. What is that? Vector? It's a normal vector to?

The level surface. Level surfaces. To a specific level surface?

No! This is a normal vector to a family of level surfaces of this whatever in R4. What dimension is this in?

Quickly, what dimension? This is the normal to the level surfaces in 3D. So this is the normal vector. To a family of level surfaces that needs all of them for the original, for the function that you created.

Now let's see if you can put it together right, Cy. Come on, come on, stick with me. Is this, look, look at the board, is this one of the level surfaces to which this is a normal vector?

Come on, stick with me. Is this one of the level surfaces to this thing of which this is a normal vector? Is it one of them?

Namely, it's the one that I set this equal to 4, that that point is actually on. So this thing, yeah, it's a normal vector to all of them at any point. This is the normal vector. To every level curve of this function at any point, it's given us all of them. How do we find a specific one?

Well, just plug it in. That's given to give you a normal vector for sure, because this is a normal vector to level surfaces. So when I plug in a point...

Oops. And we get this out. Whatever this is going to be, this is going to give you the specific normal vector to that specific level curve. At that specific point. And that's the magic.

That's the magic. Can you do it? Can you plug in those values?

Have you plugged in those values already? Can you please read them off the side what you got? Minus 4k.

Ready? Surface? Yep.

Pretend it's a level curve. Yep, now that's a level curve of that, level surface of that. Gradient finds normals to level surfaces. Find the gradient. This is a normal to the level curves, all of them, every level curve, every level surface of the function that you just created.

Surface? Yes. Function?

So that that's a level curve? Check. Take a gradient, and now this gradient is the normal to all the level surfaces.

Got it. Plug in your point. Now this is a normal to that specific level surface at that specific point.

You just found the normal to a surface at a point. What do you need to make a tangent plane? Point.

A point. Do you have a point? And a normal. Did you find a normal? Yes.

You need a point in the gradient. That's it. That's the whole shebang.

We did all of this stuff to get to this point right here. So if you want to find a tangent plane, the process is actually really easy. It's one more step after this.

Do you... Do you have a point? Do you have a normal? A specific normal that just happens to be at that point.

You've got a point, you've got a normal, you've got a tangent point. Also, you've got a point, you've got a normal, you've got a normal line. Lines need a vector and a point. Planes need a normal vector and a point.

So tangent planes, normal lines, seriously just one more pretty easy step. A little bit of review, which would be probably good for your test if you don't remember how to make planes and lines. To make a plane, you need a normal and a point. And that plane looks like this.

You put your vector stuff first, point stuff second, vector stuff first, point stuff second, components first, coordinates second. That's a plane. That's how you find any plane. Hey, hey, just make a scene again.

What gives us this one? What gives us this? A gradient.

What gives us normals? A gradient. What gives us points?

It's going to be given to you. All you've got to worry about is finding the gradient and putting a point. Done. That's it. Then, of course, you're going to solve this.

You may put it in, take that and put it in, like, standard form. Let's walk through this one last time. Lines need a point and a vector, yeah?

Points can be given to you. What type of a vector does a normal line need? You're going to use the same vector twice. The same vector for a tangent, because you need a normal, but then the normal is also the vector for the normal line.

Does that make sense to you? If you need a normal for a tangent plane and you want to put a line in the direction of that normal, that normal vector is the vector that we want. So for the normal line, you keep the same exact end.

You even keep the same point. But we have just a little different way that we do it. That's one way that we can do it. That's the, what type of equation of a line right now?

What is that? Parametric or symmetric? Symmetric. That's symmetric, that's right. Notice something.

Please, please look. Do this first every time. Here's why. If I give you the plane, if you give me the plane, if you give me that equation as the plane, from right here, do you see that you can just put this over this line, this over this line, this over this line?

It's so easy to get from a tangent plane to a normal line. It's super, super fast. If you prefer the parametric, of course, we can solve these, set them all equal to t, and solve it. And we have the parametric. That's it.

I'll tell you what, we're going to do three examples. We are going to be, maybe four if we have time, then that's it. And we're actually going to go really, really fast. So I'm going to get through as many examples as I possibly can for you guys.

Are you with me? Are you ready to follow me pretty quickly? Do your brains hurt?

No more than usual? Oh good, alright, then we're good. So for the rest of our problems we are going to find tangent planes and normal lines for all of them. It's going to be super fun and a good time.

You wouldn't rather be doing anything else the day before spring break, would you? I mean, come on. Do you want us to meet during spring break?

Yeah, sure. We could have an Easter egg hunt, and inside we'll have little math problems. Yes.

And candy. Okay, let's do it. Let's do it. Here's number one.

Number one, number one, here we go. This is a surface in R3. Look at the board.

That's a surface in R3. Let's create something in R4. Take all of your variables, put them on one side, and create a function equal to that variable side. Do you need the constants?

No. Can you all tell me what function is this going to be equal to? That's it. That's it. Can you please notice if I plug this in, that's going to give me negative 4. Do you see it?

Look at it. It gives you negative 4. It's saying this point is on that level surface of that function. I do want to point something out to you.

Some people really love to... No, no, you have to do this. X, Y, Z minus, I'm sorry, plus 4 equals 0. And then your function X, Y, Z equals X, Y, Z plus 4. You go, that has to be. That has to be it. What that does, that puts the level curve or level surface kind of at the origin, but it doesn't matter, because if you still plug this in, you get negative four plus four equals zero.

And if you take a gradient, you start doing derivatives, that four disappears. So if you want to put the constant in there, you can, but you do not have to, because this is still a level surface of that function, just like if I added the four or this would still be a level surface of that function. Does that make sense? Either way is good, it really doesn't matter. What matters is you understand that you've got to go one dimension higher by creating a different function.

So that's number one. Number two, come on now, what are you gonna do? Why are we doing gradient? Because we're supposed to, because you said so, Leonard. Why are we doing the gradient?

Find the normal. Because if this is my function, the gradient is going to give me normal. to what? The level surfaces in this case, and that's one of them. So let's go ahead, let's find the gradient.

It's going to give you the normals to the level surfaces of that function. Please go ahead and find the gradient. So fans feel okay with that one? Told you we're going to move fast. Keep in your head though, do not do math for the sake of doing math.

That's a horrible way to do math. Alright? This right here is the gradient. It is the normal vector to any level. Level surface of that function you just created.

Does that make sense? This is one of the level surfaces to that function you created. Therefore, this is the normal to that. It's just at any point. What are we going to do now?

Plug in the point. That way you find the specific normal. Are you guys kidding this?

It's super cool. And it's super cool. It's kind of easy. And now if you're with me on it. Surface, yep, big function, yep.

This is the normal to any level surface of this. This is the normal to a specific level surface, namely that one because our point was on that one. We plugged in that point. It's giving us normal to that surface. exact point that's exactly what we want to show fans if you're following on this stuff it's so cool it's awesome so now we're practically done what do you need to find a tangent plane A point.

Do we have a point? And a normal vector. Do we have a normal vector?

Yes. Done. That's it.

What do you need to find a normal line? A point and a vector that happens to be normal. That's it.

You can even do stuff like this. Because we're not dealing with length or distance or anything, we can simplify. So I'm going to call this normal vector. Let's divide by 2. I could do, or divide by negative 2 if you want.

It doesn't, it's either way. Let's choose that. If we divide by 2 or multiply by 1 half, you can simplify the normal. So we got a normal. We got a point.

If we have a normal and a point, we have a plane. If we have a normal and a point to a level surface, we have a tangent plane. That's fantastic. I did a major mistake there. What should go first, 2 or negative 1?

It's always vector, I gotta make sure we're okay with this. I don't want to stall us or anything, but do you see where all this stuff came from? For sure. Okay. From here, I want to even simplify right now.

I'm going to simplify eventually. I'd go straight to my line. Because the normal line has a normal vector and the same point, we have x minus 2 over negative 1 equals y plus 1 over positive 2 equals z minus 2 over negative 1. This over this, this over this, this over this, done.

It's that quick. It's that easy. Can you follow what I'm talking about? I would prefer symmetric if I were you because you're done.

You should leave it right there. It's easy. Could you take this and make it parametric if you wanted to or if you had to?

Yes. Now, I do want you to simplify your planes. Please don't leave them. Either one of those will work.

Typically, you're going to see this one with a positive number and not with a negative number. It's the same plane, same plane. But typically, we get the one with the positive, which means that these can switch around a little bit, but it's still the same line. Look, we've gone quickly through this, but I want to make sure that this is, man, at least well explained for you guys.

Has it been? Yes. What is this plane?

It's a tangent plane. We just managed to find a tangent plane of what? That. That.

Which was a level surface of this thing. We created it to find that. That's it.

We also found the normal so that that one line that's going to be perfectly perpendicular or orthogonal to our plane and shoot right through that point, that's that line. That is what we're supposed to do. I'm going to have you try one on your own. I'm not going to give you long. These things are going to be very, very long.

are very quick. So I'm going to try one with you, maybe do like one on our own, and then we're going to be done. So, finally, the tangent plane and normal line of this 3-D-R-3 surface at this point.

As you're going through it, try your best to think through the process. I'm giving you something in 3D. I want you to create a function of just the variables, no constants, in 4D.

Then the gradient gives me something back in 3D that's the normal to all the level surfaces, this being one of them, this being one of the level surfaces. Therefore, it's the normal to that curve. If I plug in the point, it's normal to that curve at that point.

Try that. Go for it. Again, I'm going to give you maybe two things. Were you able to create your function?

Yes. Do you need the plus six? No. Let me be very specific here.

If you don't put the plus 6, it's like you subtracted this over and you have a level curve, level surface at the negative 6. Does that make sense? If you put the plus 6, it's like you have a level surface at equaling 0. Either way, it's going to work out just fine for your discipline of math. Thank you.

How do you feel about taking partial derivatives right now? Good practice, huh? You're welcome.

You're especially going to like the practice on that one right there. Trust me. For your test coming up in like 25 minutes. I love you Leonard!

Best ever! I know. It's because I bring you candy.

And knowledge. I'm going to pause you here for the sake of time, just so we can get through this here real quick. How many people were able to create the function and find the gradient? Show of hands if you were able to find that.

That's fantastic. That's really good. Now, next question.

Do you understand that this... This right here, this is the equation for every normal vector to every level surface of this crazy looking function and that level, all the family level surfaces includes this one. So this is the equation for any normal vector to that surface.

And now that you understand that. If I plug in the point, this is the normal vector to that specific level surface function at that point. And now you're going to be understanding that one. What do you need for a tangent plane? Come on, say it, say it, put it in your head.

You need what? A point. Do we have a point? Yes. And a normal vector.

Do we have a normal vector? Yes. Gradients are normal vectors. That's the magic of this section. Do I have to, right-siders only, do I have to use this as my normal vector?

No. What could I do instead? Let's do that to make it nicer.

Could you? Yeah. It's not a problem.

You can always use that one. It'll be just fine. But to save some effort, let's divide everything by three and get... just another normal vector. So big function, level surface, this is the equation gradient, is always a normal to a level whatever.

So this is the equation for the normal vectors to this or any level surface. This is at that point because we plugged it in. This is a normal. We can just shrink this by dividing by 3, and we have our point which is that. Plains Tape Normal vector component and point coordinates and put them together.

Let's check our work here. Did you get the same thing I got? You don't have these interchanged, right?

We got our vector coming first. From here, what are you doing? By the way, what is this, plane or line? Because they look really similar.

What is that? It's plane. From here, I'm going straight to my normal line. If I can.

If I wanted to, I could solve each of these for x, y, and z by setting each of these equal to t and get parametric. But symmetric is the same exact thing. You can keep it that way.

That's fine. And now if you understand, that's a normal line. So we've got this weird surface.

Don't know what it looks like. It's a weird surface. At that point, this is the line that shoots through that point that's perfectly orthogonal to a tangent plane at that point. The tangent plane itself is right here. Did anybody solve that down and get it in a little bit nicer format?

Can you say what that is for me, please? X plus three y minus four z. Yeah, equals what?

How much? Negative seven? I don't know, I didn't do it. Oh wait. Negative 7. I get negative 11. Yeah.

You get negative 11? Negative 11. I get that. Check it out.

Yes. Should the normal line of the first term be x plus? You know, I think I did that twice.

Thanks. You know how something that looks weird and you can't catch it? Like on your tests. This is the thing you look for when you check your work.

Stuff like that. You're welcome for this one. You're welcome. Again, people who are taking tests in 20 minutes. after our break.

Yeah, normal line, got it. Tangent plane, got it. How else would you see this written? Well, that could be negative, minus, plus, positive.

That could be the other way you could write that plane. Same plane, though. Same exact thing. So this, here's a surface.

This sits flat. This plane sits flat. Now listen, sometimes books try to trick you, problems try to trick you.

If you're ever given something that is explicitly defined, this is implicitly defined, this is explicitly defined. Z is solved for... in terms of x and y.

Be smart about this. We're not going to finish this problem, but we're certainly going to do this one. Check this out. Check this out.

What I've told you is if that's 3D, which it is in R3, it's dependent equals. two independent. You with me?

Get all your variables on one side. That's what we've been doing all the time. Get all your variables on one side, constant on the other, and that becomes your function.

So you have two choices here. Watch carefully. We could choose to make this that or Now, either one.

Do you guys see that they're exactly the same? Pick one. It doesn't even matter. It doesn't matter. So, if we have this equal to 4, this is now our level surface to some function higher, or this is now our level surface to some function higher.

That's what we'd be doing here. The function itself... Would be this or if you want to on your own, you can even check it out.

Check it out. If you plug in, well, I'm not giving you a point, but if you plug in a point, this would equal that number. If you plug in a point, this would equal that number. This is a level surface. to this 4D curve, this is a 3D level surface, to this 4D curve, 4D surface, something, whatever that is.

You guys understand what I'm talking about? This is what you'd find the gradient of, one of those two things. Also, you can be smart about it. You know how to take the. derivatives with ln, making this ln x minus ln y minus z, verify that you can do that, would make your work a lot easier.

That'd be pretty nice. Do stuff like that. Do stuff like that. That's so convenient to do that.

Then you find gradient, and you're practically done. Gradient is the normal to the original thing you're given if you build it one dimension higher. Let's try it with this one.

What's the first thing I'm doing on that problem? Everybody in class right now, what's the first thing I'm doing on that problem? Variables one side, left side, how would I do it?

Would you subtract z or would you subtract tangent? It doesn't matter. What did I do? I think I subtracted z. Let's go ahead.

I want you to subtract z from both sides. Do you understand the reason why? I need my variables on one side to create that bigger function.

It has to have all of our variables in it. So we get our variables on one side. Next, I want you to create the function.

Take five seconds, create the function for me. Come on, quickly, create the function. Right-siders, what's the function that you, sounds funny coming up, what's the function?

All right, what's the function that we get out of this thing? That's it. Is this super hard for you or pretty easy for you?

Pretty easy? So, get everything on one side, that's your function. Why?

Why? Why is that your function? Well, because if this is in 4D, then this right here, nothing.

This one, this one is a level surface in 3D. The level surface, namely, when I set this equal to zero. If I set a function equal to a constant, it gives me a level surface. If I set this equal to zero, it gives me that level surface.

That's the whole point. This is a level surface to that new function that you just created. Cool, huh? Now, what gives us normals to levels? What gives us normals to level surfaces?

Come on, everybody. What gives us normals to level surfaces? So when I create this function to which this is a level surface, I get the gradient.

It's the normal to the level surfaces. Find the gradient. You know what?

Now, we've actually done this before. I'm going to, just for the sake of time, give you what the gradient is. This is that one where you do chain rule.

You have chain rule with respect to x and y, and you divide by 1 plus this thing squared, and we simplify. This is, you should have to trust me on it, or maybe if you're doing this again, pause it, or if you want to look at your notes later, try it on your own. What this gives us is this.

We've actually done it before in this class. Plus this. Minus that. Partial with respect to x is this thing. Partial with respect to y is this thing.

Partial with respect to z, this is gone. It's just that thing. Gradient takes the partials, puts i, j, and k. That right there is the, you just have to trust me or do it on your own and verify that that is correct. Do you guys know how to do partial derivatives?

Yes. Okay, we don't have time to practice that right now. What is it though?

Come on, this is the big deal right now. What is this thing? Normals. Normals. Two.

Any level surface. This is one of them. But this is one of them only at that point. So we got to plug in that point to this normal vector equation to get a specific normal vector to a specific level curve at a specific point.

Let's plug that in. It looks harder than it is. There's no Z to plug in, so you don't have to deal with it. So it's negative 1 half i here, negative 1 half j, and then this doesn't change, so minus k. You still with me?

Brains hurt more? Yes. Don't worry. You'll be fine.

You'll be all right. What is that? A specific normal vector.

A specific, I love it, a specific normal vector to that specific level surface at that specific point. What do you need for a tangent plane? Point normal.

Point and? Normal vector. We got it. Would you use that normal vector? Say it again.

Multiply everything. Let's multiply everything by 2. You can do that. Scalar multiples, you're just picking a normal.

As I'm writing down the last example we're going to do in this section, I want you to find the tangent plane and normal line. Thank you. Let's just make sure the numbers are in the right spot and I'll be pretty happy here.

Did you all get the negative 1, 1, negative 2, minus 1, minus 1, minus pi before? Hopefully, you did not, you did. From here, go straight to your normal line and you're done.

After that, you might simplify it. Did anybody simplify that? Can you tell me what you got, maybe?

I just might have to swipe plus 2z. It's pi over 2. Okay. Equals how much?

Negative. Negative pi over 2. Or change your signs and have a positive pi over 2. Either one. They're both the same.

Same thing. Can you try one? Let's try one on your own. Take this as very good practice for partial derivatives, please.

This is why we are doing it right now. Can you please take this as very good practice for partial derivatives right now? Set up the main function.

Find your gradient. That could be done. Bless you.

Thank you. About every 20 seconds I'm going to put a new step up on the board. I just want you to make sure you're checking your work.

Stick with me. Did you get your major function correct? The one that's one dimension higher, did you get that correct?

Do you need that 3 in there? Could you? Yeah. Subtract it, it's fine, it doesn't matter.

It's the same thing. It's getting the level surface. Did you get your first partial derivative with respect to x? Correct.

Did you? Yes. Did I?

Yes. Okay. You know where the y came from?

Yes. What rule is that? Chain rule. What's chain rule?

Chain rule gives us x. Oh, last one, we can't miss that. Oh, maybe. How much is it? Last time I could explain it, I'm going to milk this thing for all it's worth.

That's the surface you want. That's a function that's one dimension higher. This is the equation for the normal vectors to any level surface of this one. This is a level surface of this one.

Therefore, this... is the equation for the normal vectors to that surface at any point. I want them at a specific point.

That's why I plug it in. What we're about to get is the specific normal vector to that specific level surface of this function at that exact point. Tangents need normals.

Normal lines need normals. That's why we're finding the gradient, which always gives us normals to a level surface or a level curve. Goodness, that was a lot to say. If I plug that in, I get 3i plus 3k because that's a zero and that's gone.

Did you get it? That's the normal. Let's go ahead.

Let's find an easier normal. What's an easier normal? Quickly, what's an easier normal? Scalar multiples are super nice. We know to find tangent planes.

We take vector component, x coordinate, vector component, y coordinate, vector component. Z coordinate equals 0. This whole thing gives us X plus Z equals 1. That's tangent plane. That's what we wanted. Did you find X plus Z equals 1?

It's beautiful. From here, we got normal line as well. Please do it from here.

Please don't do it from here. Please don't do that. When we have this, we go, okay, well, how are we supposed to do it?

This gives us x over 1. This right here, you can't divide by 0. This is a weird one. So what we do with this, we do this. We do z minus 1 over 1. That's true. But this is one of those ones where that y has no t in it.

The t is 0, so it's a constant y. Equals 3. And that's how we would write that. It's a comma.

It's not equals. So we have x equals z, no problem, and then we get that y equals 3. Did it make sense to you? Are you sure you're okay with this? Easy, medium, or hard compared to the stuff we've been doing? Pretty easy.

You know what gradient does now, right? Yeah.

Transcript for:Understanding Tangent Planes and Normals

Transcript for:
Understanding Tangent Planes and Normals