Transformationssatz

Description

If you want to take the integral on some weirdly shaped space, you might want to deform the shape into something nicer and take the integral there. Since some parts will get squished, we need a formula to translate between the two integrals.

This is also useful as a integration tool.

Definition

Sei $U,V\subseteq {\mathbb{R}}^2$ offen, $\Phi :U\to V$ ein $C^1$-Diffeomorphismus und $f:V\to \mathbb{C}$ messbar. Dann ist auch $g:U\to \mathbb{C},y\mapsto (f\circ \Phi )(y)\cdot |\det d{\Phi }_y|$ messbar. Weiter ist $f$ genau dann $\lambda$-integrierbar, wenn $g$ $\lambda$-integrierbar und es gilt: ${\int }_{V}f(x)d\lambda (x)={\int }_{U}g(y)d\lambda (y)={\int }_U(f\circ \Phi )(y)|\det d{\Phi }_y|d\lambda (y)$

Properties