View source for Change of variables in an integral

[[Category:Analysis]]
{{MSC|26B10}}
{{TEX|done}}

A formula which generalizes to multidimensional integrals the usual [[Integration by substitution|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|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|Jacobian determinant]] of the [[Jacobian|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|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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|valign="top"|{{Ref|IP}}||  V.A. Il'in,   E.G. Poznyak,   "Fundamentals of mathematical analysis" ,  '''1–2''' , MIR  (1982) {{MR|0687827}} {{MR|0687828}} 
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|valign="top"|{{Ref|Ku}}|| L.D. Kudryavtsev,   "Mathematical analysis" , '''1–2''' , Moscow  (1973) 
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|valign="top"|{{Ref|Sp}}|| M. Spivak,   "Calculus on manifolds" , Benjamin/Cummings  (1965) {{MR|0209411}} {{ZBL|0141.05403}}
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|}