Why do we need a tangent space? Why do we need a vector field? ("Calculus on Manifolds" by MIchael Spivak)

Question

I am reading "Calculus on Manifolds" by Michael Spivak.

Why do we need a tangent space?
Why do we need a vector field?

To be precise, a vector field is a function $F$ such that $F(p)\in\mathbb{R}_p^n$ for each $p\in\mathbb{R}^n$. For each $p$ there are numbers $F^1(p),\dots,F^n(p)$ such that $$F(p)=F^1(p)\cdot (e_1)_p+\cdots + F^n(p)\cdot (e_n)_p.$$

I think the following definition is sufficient.

A vector field is a function $F$ such that $F(p)\in\mathbb{R}^n$ for each $p\in\mathbb{R}^n$.

If we want to know a vector at $p$, we just need to calculate $F(p)\in\mathbb{R}^n$.

In fact, in introductory electromagnetism textbooks, the definition of tangent space is not provided.

Answer 1

We need a tangent space because we're trying to do calculus on manifolds, which means we want to do calculus without preferring a particular coordinate chart. When generalizing vector calculus on $\mathbb{R}^3$, many concepts that were identified become distinct. Vectors in the sense of the derivative of a path, and vectors in the sense of a gradient of a scalar function become tangent vectors and covectors. Vectors at one point are no longer directly comparable to vectors at another point, so we need a tangent space at each point.

Answer 2

It may help to think of your differentiable manifold (I'll call it $ M $) concretely as a $ 2 $-dimensional surface embedded in $ \mathbb R ^ 3 $. (This is the sort of manifold that the pioneers of the field first studied, after all.) So you have a point $ p $ on this surface $ M $, and you can see that some of the vectors (in the ordinary sense of elements of $ \mathbb R ^ 3 $) are tangent to $ M $ at $ p $, while others are normal (perpendicular), and most are neither (while the zero vector is technically both). An $ \mathbb R ^ 3 $-valued field on $ M $ maps each point $ p $ to an element of $ \mathbb R ^ 3 $, but a tangent vector field is specifically one that maps each point $ p $ to a vector that's tangent to $ M $ at $ p $.

The vectors that are tangent at $ p $ form a $ 2 $-dimensional subspace of $ \mathbb R ^ 3 $, which you can write as $ \mathrm T _ p M $ and call the tangent space to $ M $ at $ p $. (The normal vectors form a $ 1 $-dimensional subspace $ \mathrm N _ p M $, and $ \mathrm T _ p M $ and $ \mathrm N _ p M $ together span $ \mathbb R ^ 3 $.) Since this is a $ 2 $-dimensional vector space, we could assign it a basis and identify it with $ \mathbb R ^ 2 $, and I gather that this is what Spivak is doing when he writes $ \mathbb R ^ 2 _ p $. But even in this concrete setting where $ M $ is a submanifold of $ \mathbb R ^ 3 $, there's no canonical way to do this. (At best, there are $ { 3 \choose 2 } = 3 $ canonical ways, each given by picking $ 2 $ of the $ 3 $ coordinates of $ \mathbb R ^ 3 $. At any given point on the manifold $ M $, at least one of these must work, but it's possible that none of them will work everywhere on $ M $, and guaranteed that none of them will work everywhere if $ M $ is closed. Even so, it's useful to be able to use any basis.)

When we make $ M $ an abstract manifold, we keep the terminology ‘tangent vector’, but we can no longer think of these as vectors in some ambient space $ \mathbb R ^ N $. (The Whitney embedding theorem tells us that it must be possible to embed $ M $ into some $ \mathbb R ^ N $, at least as long as $ M $ is paracompact as it usually is in practice, but there's still no canonical way to do this.) So instead we can think of tangent vectors at $ p $ as derivations of smooth or differentiable functions defined on $ M $ near $ p $, as equivalence classes of smooth or differentiable curves in $ M $ through $ p $, or as compatible ways to assign elements of $ \mathbb R ^ n $ to smooth or differentiable coordinate charts on $ M $ around $ p $. This makes the whole concept more abstract, but its origins are the vectors tangent to a surface in $ \mathbb R ^ 3 $.