In the book Analysis II by C.T. Michaels the differential is introduced as the Jacobi-Matrix. In class we had the following definition:
Definition: Let $U \subset \mathbb{R}^m$ be open, $f: U \to \mathbb{R}^n$, $x_0 \in U$
$f$ is at $x_0$ differentiable $ \iff \exists A: \mathbb{R}^m \to \mathbb{R}^n$ such that $ \lim\limits_{x \to x_0\\x \neq 0} \displaystyle \tfrac{f(x)-f(x_0)-A(x-x_0)}{\lVert x-x_0\rVert}=0$
Note: We call $A \in \hom(\mathbb{R}^m, \mathbb{R}^n)$ the differential of $f$ at point $x_0$ and we also write $A=df(x_0)$
My tutor said there are many books where the Jacobi-Matrix and the differential are said to be equal, but he mentioned they really are not.
Oddly enough when I returned home after this Colloquium in Mathematics I did try to get some practice and just found the definition of the differential being equal to the Jacobi-Matrix as in C.T. Michaels.
Now consider the following exercise (found in a paper by Salamon)
Exercise: Show that $f$ is differentiable and compute the differential $df$ for all points in the domain: $$ f: \mathbb{R} \longrightarrow \mathbb{R}^2, \ f(x)= (ye^{ix}, xe^{iy}) $$
My approach: Showing that $f$ is differentiable is easy, I compute the Jacobi Matrix $$J_f= \begin{pmatrix}iye^{ix} \\ e^{iy} \end{pmatrix} \in \text{Mat}_{2,2}( \mathbb{C}) $$And see that all the partial derivatives exists and are continuous $ \implies f$ is differentiable.
Questions:
- How do I find the differential? If I plugin the Jacobi-Matrix into the definition above I can't seem to come up with the correct result
- Is it wrong to treat the Jacobi-Matrix and the differential as equal? (considering the definition as given in my class of course)