think of a curve we'll locate this curve in a dimension which we can easily imagine in two Dimensions now draw a line which lightly touches the curve at just one specific point this is a tangent line now say that instead of a line we draw a two-dimensional space much like a Loosely hanging sheet this is a manifold by the way and it is located in a three-dimensional space say we pick a point p and want to draw something equivalent to a tangent line but in two Dimensions we would therefore draw a tangent space in this case it looks like a flat plane just touching the point in exactly one place it's labeled TPM and literally means the tangent space of the manifold M at Point P mathematically we would describe the tangent space just like how we describe locations in everyday space using coordinates like X Y Z in 3D and we describe dire C in the tangent space using coordinate systems tied to the manifold on a 2d manifold we might use coordinates like X1 and X2 and these are local variables that change smoothly as we move around the manifold these coordinates help us describe the position of a point on the manifold and they also help us to describe directions at the point p in the tangent space well what's the point so what we drew a plane lightly touching a point now what well actually these tangent spaces are abstract constructions that represent all possible directions in which one can move from a point on a manifold this concept is known as a vector vectors are arrows which have Direction and magnitude much like going and then all of a sudden stopping and noting how fast you were walking and in what direction on a curved line there are only two possible directions for vectors forward and backward along the curve on a two-dimensional tangent plane there are infinitely many possible directions forming a circle of choices or directions around the point P they are important because they describe how steeply a curve climbs or Falls any point when we have a higher dimensional manifold embedded in a higher dimensional ukian space these Vector spaces become more abstract and impossible to visualize these spaces still touch the manifold at one specific point but here's the catch manifolds don't actually need to be embedded in a higher dimensional space they kind of exist on their own we study them abstractly independent of any embedding let's see if you got that a manifold does not have to have a space surrounding it it is the space and there is nothing Beyond it that's why vectors can't poke out of it because there's nothing to poke out into instead to find vectors and perform calculus you have to pick a point and localize it or conceptually separated from the manifold itself this is similar to zooming in on a globe of the earth until it looks flat except the Earth is the only thing that exists there's no space surrounding it that's what we are doing conceptually when we use charts to map part of a curved manifold onto ukan space where we can perform calculus so again we can only properly measure things when we embed a portion of the manifold to locally resemble a ukian space so the question becomes how are we supposed to do that and how would you map an entire tangent space first let's start with finding out just one single Vector we have a manifold M and pick a point P since we are talking about a manifold in higher Dimensions it's pretty hard to pinpoint the exact location of the P so instead we create a neighborhood points where P floats freely but does not exit the boundaries of the region and label it U it is formally called an open subset of M in order to perform any kind of calculus at all on P and on U we have to map it to a ukian space RN where Calculus is well defined this is done using a chart fi this chart assigns local coordinates that describe the point on M near P these coordinates let us flatten the manifold onto ukan space where we can do calculus so say we have a curve gamma passing through the manifold through the point P here we also have to map the parameter of time or t remember we need to find the velocity of the vector which includes the component of time not only where it finds itself geographically T which finds itself in another ukian space r the line itself is just one dimensional representing time T takes values in the interval a in the real line and therefore we map t to the manifold M using gamma the parameter T tells us where we are on the curve at a specific time for example gamma of t0 equals P which means that at time t0 the curve passes through the point P to calculate the velocity of the curve at P we need to work in a ukle in space RN where derivatives are well defined to do this we combine the chart fi and the curve gamma of T the composition F composed with gamma Maps the curve on the manifold to a curve in RN this composition effectively relocates the curve from the manifold into the ukian space allowing us to perform calculus on it once the curve is mapped to ukian space we can finally compute the velocity vector at Point P by taking the derivative of f composed with gamma of t with respect to T this derivative gives the velocity Vector of the curve at the point P equals to gamma of t0 but now expressed in terms of the local coordinates all of this was to introduce tangent vectors as velocities of Curves passing through a point p on the manifold but this is not enough for practical analysis because a single tangent Vector is just one direction of movement at P but instead we need to describe all possible directions at P which forms the tangent space of the manifold M at Point P if you're enjoying this video please don't forget to like it and to subscribe to the channel to study the manifold globally so not only in one specific point we also need a way to compare different tangent spaces at different points or under different charts this is where the idea of a basis comes in so we want to be able to do analysis between points and charts not just a single pointer chart in order to do so we need to introduce an arbitrary function f an entirely different field called a scalar field what it does is assigns scalars to each point on M scalers are much like vectors except they don't have a direction it's much like velocity versus temperature for example it's like if we pick a point p on the manifold and apply F we will therefore assign it a number or a scaler just for the sake of intuition let's say we're assigning temperature again just as an example now say we want to find the rate of change of our function f as we walk along our curve gamma of T we want to look at a new definition of velocity relative to this Tas function at our Point p on the manifold but here's a really fun trick we can do by introducing fi and its inverse it's a trick because it doesn't really do anything it's just like adding zero or multiplying by one because this introduction is quote unquote harmless we can therefore put it like this now we introduce the chain rule which we won't go into detail but if you want to know more let us know in the comment section below which brings us to here since P equals gamma of t0 with the left part being the basis for component I and the right part the veloc vity of component I with respect to F here's something that is a little weird we're going to rewrite the basis like this we see that F disappeared but it's just a convention and five specified by XI and what is the F it was an arbitrary test function we just needed it in the calculations to get us where we got to you can pick literally any different differentiable function f it doesn't really matter what it is we only needed it to define the basis of the tangent space so it was so arbitrary that we can actually get rid of it now so for every tangent vector v in TPM we have this it turns out the basis is actually a set of differential operators not the actual vectors on the test function f which make up a vector space a vector space doesn't need to be our usual ukian vectors they can be anything that satisfies the vector space properties including differential operators it is important to remember that there is a chart behind it or under different charts so we found a basis to the tangent space at a specific point and since we want to do it for every point in the manifold the next logical step would be to compare different tangent vectors in different tangent space at different points in the manifold M so if if you're curious to know more leave us a comment also do not forget that we include a PDF which is like a summary of everything we see in the video this way you guys can study by yourself at home and remember that's the only way to really learn math by understanding each step each detail and then trying to reproduce everything independently by yourself I guarantee you that it's going to be very beneficial so download the PDF Link in the description below this video was inspired by this article and this book Link in the description also if you like this video I'm pretty sure you're going to love this one see you guys there