Change of variables in an integral

2020 Mathematics Subject Classification: Primary: 26B10 [MSN][ZBL]

A formula which generalizes to multidimensional integrals the usual integration by substitution of integrals in one variable.

Let $U$ and $V$ be open sets in $\mathbb R^n$, $\Phi: U \to V$ be a diffeomorphism and $f: V \to \mathbb R$ a continuous function. For any $y\in U$ denote by $J \Phi (y)$ the absolute value of the Jacobian determinant of the Jacobian matrix $D\Phi|_y$, i.e. the determinant of the $n\times n$ matrix \begin{equation}\label{e:Jacobi_matrix} D\Phi|_y := \left( \begin{array}{llll} \frac{\partial \Phi^1}{\partial x_1} (y) & \frac{\partial \Phi^1}{\partial x_2} (y)&\qquad \ldots \qquad & \frac{\partial \Phi^1}{\partial x_n} (y)\\ \frac{\partial \Phi^2}{\partial x_1} (y) & \frac{\partial \Phi^2}{\partial x_2} (y)&\qquad \ldots \qquad & \frac{\partial \Phi^2}{\partial x_n} (y)\\ \\ \vdots & \vdots & &\vdots\\ \\ \frac{\partial \Phi^n}{\partial x_1} (y) & \frac{\partial \Phi^n}{\partial x_2} (y)&\qquad \ldots \qquad & \frac{\partial \Phi^n}{\partial x_n} (y) \end{array}\right)\, , \end{equation} where $\Phi^1, \ldots , \Phi^n$ denote the components of the vector function $\Phi$.

Then the following formula holds for any compact $\Omega\subset U$: \begin{equation}\label{e:change_of_variables} \int_\Omega f (\Phi (y)) J \Phi (y)\, dy = \int_{\Phi (\Omega)} f (z)\, dz\, . \end{equation}

Formula \eqref{e:change_of_variables} plays a fundamental role in defining the integration of a differential form: see also Integration on manifolds.

The assumptions on $\Phi$, $f$ and the domains can be relaxed in several ways: we refer to Area formula.

References

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