Let $f : \mathbb R
^2 → \mathbb R
^2$ be a differentiable function and denote by
$J_f$ its Jacobian matrix.
Let $γ : \mathbb R → \mathbb R
^2$ be a smooth curve with
velocity vector $γ
'$ such that $γ(0) = p$,
and let $γ_1 = f ◦ γ$ be the image
of the curve by $f$. Show that the velocity vector of $γ_1$ at $f(p)$ is $J_{f_(p)}$·$γ'(0)$
(Hint: use the chain rule for partial derivatives)
I've come across this problem and I don't know how to solve it. I just started introduction to complex analysis and I'm struggling. Can anyone please help me with the solution?
Best Answer
If $f,g$ are suitably differentiable then the chain rule gives ${\partial (f \circ g)(x) \over \partial x } = {\partial f(g(x)) \over \partial x } {\partial g(x) \over \partial x}$
Here $\gamma_1 = f \circ \gamma$ so with a simple symbol substitution you get $\gamma_1'(0) = {\partial f(\gamma(0)) \over \partial x } {\partial \gamma(0) \over \partial x} = J_f(p) \gamma'(0)$.