What does it mean by "The differential of a map is independent of coordinate charts"?

Question

We know that the differential is represented by the Jacobian matrix $[\partial F^i/\partial x^j(p)]$, so it definitely depends on the choice of coordinates $x^j$. But the statement in question is written in Tu's introduction to smooth manifolds in the discussion of regular level set theorem. Can somebody please explain what that statement mean?

Answer 1

Quick answer:

The differential of a map associate a vector $df[v]$ to any vector $[v]$, and that vector $df[v]$ depends only on $v$, not on the coordinate system.

Detailed discussion:

What is a map? A map $f:M\to \mathbb R^n$ associates to any point $x\in M$ a vector $f(x)$. The value $f(x)$ depends only on the point, not on local coordinates.

What is the differential of a map? The differential of a map is not a function from $M$ to $\mathbb R^n$. One usual way to define the differential of a function $f:M\to \mathbb R^n$ is the following:

$f$ is differentiable at $x\in M$ if there exists a linear map $L$ so that $$f(x+\epsilon v)=f(x)+\epsilon L(v)+o(\epsilon)$$

In other words, if $f$ is approximable by a linear map at first order. So, now look at $L$. Who is $L$? It is a linear map bur from where to where? The origin vector space is the tangent space of $M$ at point $x$, usually denoted $T_xM$, and the target space is the tanget space at $f(x)$. In the present case, we can canonically identity the tangent space at $f(x)$ with $\mathbb R^n$ (but if you have $f:M\to N$ you have to take $T_{f(x)}N$). So $$L\in\hom(T_xM,\mathbb R^n)$$.

The map $L$ is called the differential of $f$ at $x$ and it is usually denoted by $d_xf$.

Now, it is a general linear algebra fact that if you have two finite dimensional vector spaces $V,W$ then, for any choices of basis $B_V,B_W$ of $V$ and $W$, you can associate to any $F\in\hom(V,W)$ the matrix of $F$ in those chosen basis. The linear map $F$ exists independently on $B_V,B_W$: it is the matrix that changes when you change coordinates.

Well, for $df$ is the same: the linear map $d_xf:T_xM\to \mathbb R^n$ does not depend on coordinates, the Jacobian matrix --- which describes $d_xf$ as matrix --- does indeed depend on coordinates.