welcome to all of calculus 3 in 8 minutes disclaimer I cannot fit absolutely everything in here as it would make the video way too long thanks to fuzzy penguin Ams for inspiring this video check out their video on calc 2 which I've Linked In the description now let's get into it part one 3D space vectors and surfaces just like in 2D there exist functions in 3D these functions take two inputs on the X and Y axes and produce an output on the z-axis vectors or quantities with Direction and magnitude this also applies to 3D vectors just their direction is three-dimensional part two Vector multiplication we can do operations like addition and subtraction with vectors just like one would with regular numbers what's different with vectors though is that there are different types of multiplication the first is called the dot product and it's the most intuitive just multiply each of the numbers in a vector by the corresponding one in the other vector add the results up and that's your dot product this may seem arbitrary but there exists an interesting connection between dot products in the spatial relationship between two vectors two perpendicular or orthogonal as they're called in 3D space vectors we'll always have a DOT product of zero for two parallel vectors the dot product is equal to the product of the magnitude of the two vectors this property extends to become the following formula where Theta is the angle between the vectors u and v and the double absolute values mean magnitude of the cross product is the other main way of multiplying vectors this product can be expressed as the determinant of this Matrix where I J and K are the unit vectors in each of the X Y and Z directions respectively and V and U are the vector being multiplied the cross product also has an interesting property it always generates a vector that is orthogonal or perpendicular to the two vectors being multiplied part three limits and derivatives of multivariable functions just like with 2D functions where we approach an x value we can take the limit of a 3D function where we approach a value on the X Y plane using these limits we can also Define a derivative just like one would in 2D one thing to note is that there are infinitely many derivatives of a 3D function at a certain point since there are infinitely many directions one can go in each of these are called directional derivatives also taking the partial derivative with respect to X gives us the derivative in the positive X Direction and differentiating with respect to Y gives us the derivative and the positive y direction the gradient is a generalization of this gradient and a point is a vector with components being the derivative in the X Direction and Y Direction part four double integrals just like we Define differentiation integrals also exist with multivariable functions when doing a double integral instead of finding the area under a function we're finding the volume under it to do this first do an integral in the X Direction and then integrate that area a second time over the y direction effectively extruding it but that isn't all we can do with double integrals you can also integrate over non-rectangular region instead of having the bounds of the integrals be numbers they can be functions we can also use polar coordinates for double integrals where instead of making an integral d y d x it is Dr D Theta notice that the polar equation we're integrating is R cubed instead of r squared more on that later part 5 triple integrals and 3D coordinate systems triple integrals can also be used for volume triple integrals we can integrate any 3D region as long as we can Define it with three different boundaries instead of two here x goes from 0 to 2 y goes from 0 to X forming a triangular base and Z goes from zero to Y squared creating a sort of parabolic shape triple integrals can also be used for finding things like average temperature where we would integrate the temperature function over a 3D surface to find the average temperature when doing a double integral sometimes it's easier to convert to Polar coordinates likewise there's an analogous system in 3D space or rather two analogous systems cylindrical and spherical coordinates in cylindrical coordinates there is a radius r a z coordinate and an angle Theta in spherical coordinates there is a radius and two angles rho is the 3D distance from the origin Phi is the angle from the vertical line which passes through the origin and Theta works just like in polar part 6 coordinate transformations in the Jacobian there's actually an infinite number of coordinate systems not just the ones described previously any coordinate system can be defined as x equals g of u v and y equals h of UV as an example in polar coordinates this is x equals R cosine Theta and Y equals R sine Theta when changing an integral from one coordinate system to another one must add a certain function J the Jacobian to the inside of the integral to account for the Distortion that switching coordinate systems creates let's do this for the example of a double integral shown earlier here we can see the Jacobian being added to the inside of the integral what I just said is basically an oversimplification of the reason behind the Jacobian and the real reason involves a ton of Cool Math with matrices where the Jacobian is indeterminant but that could be an entire video of its own anyways here are the Transformations needed to convert to Polar cylindrical and spherical coordinates part 7 Vector Fields scalar fields and line integrals a vector field is an assignment of a vector to each point in a plane or space a scalar field is essentially that but with regular numbers AKA scalars think of these as 3D surfaces above a 2d plane in a way they're essentially the same thing as 3D functions when you take an integral in calc one the integral is always done on a flat surface think a sheet of paper what if this sheet of paper was bent in 3D space this is a scalar line integral these integrals are done over a scalar field and yes there are also Vector line integrals done over Vector Fields these can be hard to illustrate though but you can think of them as the work done on a particle as it travels along a vector field of a force Vector Fields can be conservative meaning that any line integral on them is path independent this is very analogous to conservative forces I.E gravity or the work done on particles is path independent and in fact gravity has a vector field although this field varies depending on the objects that are present Vector Fields have two main properties Divergence and curl Divergence is the amount of outflow from a certain part of a vector field in the center the Divergence is high since everything is moving outwards well further away the Divergence is lower since some points are moving in and some are moving out curl is the amount of rotation around a certain point in a vector field here on the outside the curl is low since there is not much spin but at the center the curl is high since everything spins around it and that's it once again I left out a lot of stuff as after all you aren't supposed to learn this course in eight minutes anyways I also may make some videos later on going more in depth on the various subjects discussed here I know there are all sorts of amazing resources online if you want to go more in depth yourself I'll link some of them in the description but for now that's all see you next time