Let $\phi:\mathbb R^2 \to \mathbb C$ be the map $\phi(x,y)=z$ where $z=x+iy$.Let $f:\mathbb C\to \mathbb C$ be the function that is $f(z)=z^2$and $F=\phi^{-1}f\phi $..
Represent the derivative of $F$ at $(x,y)$ by a linear transformation.
Can someone please help me to understand what is being asked in the problem i.e what I will hve to find here?
$\phi(x,y) = x+iy$, so $f(\phi(x,y)) = (x+iy)^2 = x^2-y^2+2 xy i$.
All $\phi^{-1}$ does is to map $x+iy$ to $(x,y)$, so we have $F(x,y) = \phi^{-1}(f(\phi(x,y))) = (x^2-y^2, 2 xy)^T$.
Now differentiate $F$. The resulting matrix is the linear transformation.
I am guessing that the problem is trying to show the connection to the Cauchy Riemann equations.