Following my previous question on the meaning of dx, and my read on A geometric approach of differential forms, I was trying to work out the intuition on paper.
tldr of the post : $\frac{\partial f}{\partial x}$ the tangent vector can be thought as a line segment. $dx$ can be though as evaluating the length of a vector (i.e line segment) and takes the value 1 on the basis vector.
So let's say we want to generalize integration on manifolds, in the nice case of a Riemannian manifold with euclidean structure, we chop the coordinates axis, evaluate the volume of the little pieces and multiply that by the value of the function. So for the 1D case : \begin{align} \sum f(x_i)\Delta_i \end{align}
with $\Delta_i = x_{i+1} - x_i$. The integral is taken by taking the limit when $\Delta_i \rightarrow 0$ This is pretty intuitive and geometric.
So on a manifold (let's say we are working on a local patch and do not worry of partition of unity etc...) we have no way to chop up the manifold into little pieces to compute the volume. So we decide to approximate the manifold by a linear space known as the Tangent space $T_pM$. Elements of $T_pM$ are vectors which are directional derivative operators. I know that when you take the directional derivative of a function you find the linear approximation of the function. So I do not understand why $\frac{\partial}{\partial x}$ would be this object since no function are involve.
I understand the object $\frac{\partial}{\partial x}$ as a basis vector of the Tangent space, but not as a representative of the linear approximation of the manifold. This is my number (1) issue.
Let's say now that elements of the tangent space are indeed locally (using an euclidean structure which is always possible on a manifold locally ) represent line segments. Then we have no notion of length in $T_pM$ so we introduce differential forms that eat a Vector and spit out a number. We call this number the "length" of the vector. This work out good because everything is linear as we expect. Just the value is completely arbitrary (but at least is consistent with addition and scaling).
So looking at the littles $\Delta_i$ from the beginning we can replace them by $\omega(X_i)$ where $\omega$ is a volume form and $X_i$ a Tangent vector at the point $i$. This way we form the sum :
\begin{align} \sum f(x_i)\omega(X_i) \end{align}
We can incorporate the function f with the volume form to form a differential 1 form (using $dx$ as basis) and we have :
\begin{align} \sum f(x_i)dx_i(X_i) \end{align}
Now comes my number (2) issue. I know that $dx_i(X_i) = 1$ by definition, but the goal of the Rieamann sum is to make something goes to 0 to have a better and better approximation of the sum. So My guess is that $dx_i(X_i)$ has to go to $0$. So here is my way to do it.
We can multiply $X_i$ by $\frac{1}{n}$ and make $n\rightarrow \infty$. So :
\begin{align} \sum f(x_i)dx_i(\frac{1}{n}X_i) = \sum \frac{1}{n}f(x_i)dx_i(X_i) \end{align}
Now this last expression os attractive but I do not know how to proceed. We can always cancel $dx_i(X_i) = 1$ but then we are left with what seems to be ill-defined under change of coordinates.
So to sum up :
- Why $X_p \in T_p M$ can be seen as a line element ? It is an operator and when we plug in a function, it is the best linear approximation of the function at $p$. How come we can interpret it as being the best linear approximation of the manifold ( also supported by the drawing of an arrow attached to $p$ pointing in the Tangent direction of the manifold )
- How to derive a well defined Riemann sum for arbitrary manifold (since we can't integrate function on manifold) and how all of this fits together (intuition of volume, differential forms, and integration using Sums)
I appreciate any suggestions / references !
