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Transformation of the Region of Integration

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Let $ f$ be a continuous scalar function. A bijective, continuously differentiable transformation $ g$ of a regular region $ U \subseteq \mathbb{R}^n$ with

$\displaystyle \operatorname{det}g^\prime(x)\ne 0,\quad x\in U \,, $

has the following influence on the multiple integral of $ f$ over $ V = g(U) $ :

$\displaystyle \int\limits_U f\circ g\, \vert\operatorname{det}g^\prime\vert\,dU = \int\limits_V f\,dV\, \ , $

where $ \operatorname{det} g'$ is the jacobian determinant of the transformation. It describes the local change of the volume element

$\displaystyle d V = \vert\operatorname{det} g'\vert \, dU \,. $

\includegraphics[width=\moimagesize]{a_trans_mehr_3}

For a local orthogonal coordinate transformation $ g$, i.e. the columns of $ g^\prime$ are orthogonal, the jacobian determinant has the form

$\displaystyle \vert\operatorname{det}g^\prime\vert = \prod_{i=1}^n \left\vert \frac{\partial g}{\partial x_i} \right\vert \,, $

i.e. the scaling factor of the volume is the product of the scaling factors of the several variables.

The conditions can be formulated weaker, e.g. it suffices to require the bijectivity of $ g$ and the invertibility of $ g^\prime$ in the interior of $ U$. Also if both integrals exist $ f$ may have some singularities.

Examples:

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  automatically generated 5/30/2011