Problem 1542 (difficulty: 5/10)

Let \(\displaystyle f\) be regular on the domain \(\displaystyle D\) with image \(\displaystyle D'\). Assume that \(\displaystyle f\) is injective and let the area of \(\displaystyle D'\) be \(\displaystyle A(D')\).

(a) Prove that

\(\displaystyle A(D')=\displaystyle\int\limits_D |f'(z)|^2\dx\dy. \)

(b) Compare with the theorem on \(\displaystyle \RR^2\to\RR^2\) functions.

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