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Change of variables in an integral

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2010 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

[EG] L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. MR1158660 Zbl 0804.2800
[IP] V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 1–2 , MIR (1982) MR0687827 MR0687828
[Ku] L.D. Kudryavtsev, "Mathematical analysis" , 1–2 , Moscow (1973)
[Ni] S.M. Nikol'skii, "A course of mathematical analysis" , 2 , MIR (1977) MR0796320Zbl 0479.00001
[Ru] W. Rudin, "Principles of mathematical analysis", Third edition, McGraw-Hill (1976) MR038502 Zbl 0346.2600
[Sp] M. Spivak, "Calculus on manifolds" , Benjamin/Cummings (1965) MR0209411 Zbl 0141.05403
How to Cite This Entry:
Change of variables in an integral. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Change_of_variables_in_an_integral&oldid=29207