Possible definition of Nabla useful? I am working on the following exercise:
Let X be an n-dimensional differentiable manifold, and $f : X \longrightarrow R$ a differentiable function.
One could try to define the gradient $\nabla f (p) \in T_p^{alg} X$ of f at p as follows:
Choose a chart (U,h,V) for X around p with coordinates $(x_1,...,x_n)$ in V, and define
$$\nabla f(p) := \sum_{i=1}^n \frac{\partial (f \circ h^{-1})}{\partial x_i} (h(p)) \cdot \frac{\partial}{\partial x_i}(p)$$
Is that a good definition?
I ask myself: What would make a good definition? With what should it be compatible?
 A: It's a great question... it would be a bad definition by most standards for the following reason. Let $j=(y_1,\ldots,y_n)$ be a different choice of coordinates on the same region, or on an intersecting region. On the one hand, the expression you wrote down in terms of $h,x$ is (in terms of $j,y$), using $f\circ h^{-1}=(f\circ j^{-1})\circ(j\circ h^{-1})$,
$$\sum_{i=1}^n\sum_{k=1}^n\frac{\partial(f\circ j^{-1})}{\partial y^k}\Big|_{j(p)}\frac{\partial(j\circ h^{-1})^k}{\partial x^i}\Big|_{h(p)}\sum_{\ell=1}^n\frac{\partial(j\circ h^{-1})^\ell}{\partial x^i}\Big|_{h(p)}\frac{\partial}{\partial y^\ell}\Big|_p$$
using the chain rule to arrive at the first two terms and the transformation rule for coordinate vector fields to arrive at the last two terms. The problem is that, in general, that will not equal
$$\sum_{i=1}^n\frac{\partial(f\circ j^{-1})}{\partial y^i}\Big|_{j(p)}\frac{\partial}{\partial y^i}\Big|_p$$
since this would require
$$\sum_{i=1}^n\frac{\partial(j\circ h^{-1})^k}{\partial x^i}\Big|_{h(p)}\frac{\partial(j\circ h^{-1})^\ell}{\partial x^i}\Big|_{h(p)}=\delta_{k\ell},$$
which just isn't true if $j\circ h^{-1}$ is a general diffeomorphism from one open set in $\mathbb{R}^n$ to another.
So each coordinate chart around $p$ would define $\nabla f(p)$ as an element of $T_pM$, but different charts would define it as different elements. That's bad, according to the usual standards.
You should compare this to the analogous exercise which shows that
$$df(p)=\sum_{i=1}^n\frac{\partial (f\circ h^{-1})}{\partial x^i}\Big|_{h(p)}dx^i\big|_p$$
is a good definition of $df(p)\in T_p^\ast M$.
