I have to solve the following exercise:
For a smooth map $F: M\longrightarrow N$ between manifolds, and for given $v\in T_pM$, compute the differential $dF_p(v)$ in local coordinates, using its definition as the derivation at $F(p)$ that acts on $C^\infty(N)$, to show that it is represented by the Jacobian matrix.
My problem is that I don´t know how to use the derivation at $F(p)$ to compute the differential $dF_p(v)$ in local coordinates.
Can somebody help me please ?
We take local coordinate $x_1,\dots,x_n$ in $M$ and $F:M\rightarrow N$ such that $$F(x_1,\dots,x_n)=(y_1(x_1,\dots,x_n),\cdots,y_n(x_1,\dots,x_n))$$
So, using the chain rule you obtain as $dF$ works: $$ \frac{\partial }{\partial x_i}=\frac{\partial y_1}{\partial x_i}\frac{\partial}{\partial y_1}+\cdots +\frac{\partial y_n}{\partial x_i}\frac{\partial}{\partial y_n}$$
So let $v\in T_pM$, for some $\lambda_1,\cdots,\lambda_n\in\mathbb{R}, $ you could write $v=\lambda_1 \frac{\partial }{\partial x_1}+\cdots+\lambda_n \frac{\partial }{\partial x_n}$
Replacing the $\frac{\partial }{\partial x_1}$ you obtain that: $$v=\lambda_1 \left(\frac{\partial y_1}{\partial x_1}\frac{\partial}{\partial y_1}+\cdots +\frac{\partial y_n}{\partial x_1}\frac{\partial}{\partial y_n}\right)+\cdots+\lambda_n \left(\frac{\partial y_1}{\partial x_n}\frac{\partial}{\partial y_1}+\cdots +\frac{\partial y_n}{\partial x_n}\frac{\partial}{\partial y_n}\right) $$ and reordering the indexes:
$$v= \left(\lambda_1\frac{\partial y_1}{\partial x_1}+\cdots+\lambda_n\frac{\partial y_1}{\partial x_n}\right)\frac{\partial}{\partial y_1}+\cdots +\left(\lambda_1\frac{\partial y_n}{\partial x_1}+\cdots+\lambda_n\frac{\partial y_n}{\partial x_n}\right)\frac{\partial}{\partial y_n}$$
So you obtain the $dF$ map andyou could see that it works as the product with the Jacobian matrix:
$$dF(v)=J \cdot v $$
