Let \(f:\mathbb{R}^2\to \mathbb{R}^2\) (or \(\mathbb{R}^2 \to \mathbb{R}\)).
Define in terms of linear transformations what it means for \(f\) to be differentiable at a point \((a, b) \in \mathbb{R}^2\).
State a version of the inverse function theorem in this setting.
Identify \(\mathbb{R}^2\) with \(\mathbb{C}\) and give a necessary and sufficient condition for a real-differentiable function at \((a, b)\) to be complex differentiable at the point \(a+ib\).
Solution
1. Let \(p=(a,b)\). A map \(f:U\subset\mathbb R^2\to\mathbb R^m\) is differentiable at \(p\) if there is a linear map\[ L:\mathbb R^2\to\mathbb R^m \]such that\[ \frac{\|f(p+h)-f(p)-Lh\|}{\|h\|}\longrightarrow0 \qquad(h\to0). \]The map \(L\) is the Fréchet derivative \(Df(p)\).
2. If \(f:U\to\mathbb R^2\) is \(C^1\) near \(p\) and \(Df(p)\) is invertible, then there are neighborhoods \(V\) of \(p\) and \(W\) of \(f(p)\) such that\[ f|_V:V\to W \]is a \(C^1\) diffeomorphism. Moreover\[ D(f|_V)^{-1}(f(p))=(Df(p))^{-1}. \]
3. Identify \(\mathbb R^2\) with \(\mathbb C\) and write\[ f=u+iv. \]Assume \(f\) is real-differentiable at \((a,b)\). Then \(f\) is complex differentiable at \(z_0=a+ib\) iff the real-linear map \(Df(a,b)\) is actually complex-linear. A real-linear map is complex-linear iff its matrix has the form\[ \begin{pmatrix}\alpha&-\beta\\ \beta&\alpha\end{pmatrix}. \]Since\[ Df(a,b)= \begin{pmatrix}u_x&u_y\\v_x&v_y\end{pmatrix}, \]this is equivalent to the Cauchy–Riemann equations\[ u_x=v_y,\qquad u_y=-v_x \]at \((a,b)\). In that case\[ f'(z_0)=u_x(a,b)+i\,v_x(a,b). \]