tangent vs cotangent space and the derivative: intuition and an example

Question

I'm working through the first chapter of "An Introduction to Manifolds" by Tu and am drowning under formalism without a good sense of what is motivating the many definitions. I think a lot of my confusion stems from the fact that, for calculus on $\mathbb{R}^n$, I'm not used to keeping track of "where objects live" (point in manifold vs vector in tangent space $T_p(\mathbb{R}^n)$ vs vector in cotangent space, etc.). A specific point of confusion is the following:

Recall that a map $f:A\subset \mathbb{R}^n \to \mathbb{R}$ is differentiable at $x_0\in A$ if there exists linear function $\mathbf{D}f(x_0):\mathbb{R}^n\to\mathbb{R}$, denoted the derivative of $f$ at $x_0$, such that $$\lim_{x\to x_0} \frac{ f(x)-f(x_0)-\mathbf{D}f(x_0)\mathbf{(x-x_0)}}{x-x_0}=0.$$

I want to interpret this in the language of manifolds; specifically, where does the boldface $\mathbf{(x-x_0)}$ live? I know it lives in $\mathbb{R}^n$, but is it $\mathbb{R}^n$ as a manifold, tangent space, or cotangent space? From what I've read, the derivative in the context of differential geometry is a linear map between tangent spaces, suggesting that $\mathbf{(x-x_0)}\in T_p(\mathbb{R}^n)$. But this doesn't really make since to me since I conceptualize $\mathbf{x-x_0}$ as corresponding to an infinitesimal change in $x$ in limit (I know that's not rigorous), which I believe is formalized in the context of diff. geom. as the differential form $dx$, which lives in the cotangent space.

What am I missing? How exactly would one recast the definition of derivative found in a multivariate calculus book into one using the language of tangent spaces, differentials, etc.?

Answer 1

$x - x_0$ denotes, in the limit, an infinitesimal change in a particular direction, i.e. a tangent vector at $x_0$. An "infinitesimal change to $x$ in limit", as you describe it, is one way to think of what a tangent vector is.

In general, you can recognise that something is a tangent vector if it is an input to $D(f)(x_0)$, also written as $D_{x_0}(f)$. This is the differential map, so its inputs are tangent vectors, as are its outputs.

Answer 2

I see that you are confused about some of the fundamental concepts, so let me recap. A map between manifolds is said to be differentiable if it induces a differentiable map between the corresponding coordinate charts (which is basically a map from $\mathbb{R}^n$ to $\mathbb{R}^m$ so the usual definition of differentiability applies here). If you want to consider the linear map $Df(x_0)$ in the case of manifolds, then you will have to pick an interpretation of the tangent space of a manifold at a point. You can find multiple ways of defining the tangent space here. The idea is that instead of thinking of $x-x_0$ as we do in $\mathbb{R}^n$, one should consider the tangent vectors of a manifold at a point as equivalence classes of curves. This is the conceptualization of the “infinitesimal change” that you are looking for. Then the linear map, which is called the differential and should remind you the Jacobian, simply pushes those curves forward to create new vectors in the tangent space of the codomain of the map.