Differentials on manifolds

Question

I have been thinking about the problem of establishing a diffeomorphism between an open domain $A\subset\mathbb{R}^n$ and the graph of a differentiable function $f:A\to\mathbb{R}$, that is, the set $$G(f)=\{(x,f(x))\in\mathbb{R}^n\times \mathbb{R}:x\in A\}.$$ I have thought about the idea of defining a function $F:A\to G(f)$ and prove that this function is a embedding, which will give me the desire diffeomorfism between $A$ and the image of $F(A)=G(f)$. Calculating the Jacobian of $F$ it can be confirmed that F is differentiable in the classical way but the problem is that for the embedding I need to see that the differential of $F$, that is $dF_p$, is injective but I don't know how to calculate it. Can anyone help me?

Answer 1

Let $\pi: G(f) \to A$ the projection, where $G(f) = A \times f(A).$

We have $\pi \circ F = Id.$

If $dF(p).v = 0$ then $$ v = Id(v) = d(\pi \circ F)(p).v = d\pi(F(p)) \circ dF(p).v = 0.$$

Thus $dF(p)$ is injective.