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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Change of Variables in Multiple Integrals">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Sun Mar 30 14:12:50 1997}
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\section{Change of Variables in Multiple Integrals\label%
{Change of Variables in Multiple Integrals}}

\subsubsection{The Jacobian of the Transformation\label%
{The Jacobian of the Transformation}}

\begin{definition}
The \textbf{Jacobian} of the transformation $T$ given by $x=g(u,v)$ and $%
y=h(u,v)$ is 
\[
\dfrac{\partial (x,y)}{\partial (u,v)}=\left| 
\begin{array}{ll}
\dfrac{\partial x}{\partial u} & \dfrac{\partial x}{\partial v} \\ 
\dfrac{\partial y}{\partial u} & \dfrac{\partial y}{\partial v}
\end{array}
\right| =\dfrac{\partial x}{\partial u}\dfrac{\partial y}{\partial v}-\dfrac{%
\partial x}{\partial v}\dfrac{\partial y}{\partial u}
\]
\end{definition}

\subsubsection{Change of Variables in a Double Integral\label%
{Change of Variables in a Double Integral}}

Suppose that $T$ is a one-to-one $C^{1}$ transformation whose Jacobian is
nonzero and that maps a region $S$ in the $uv$-plane onto a region $R$ in
the $xy$-plane. Suppose that $f$ is continuous on $R$ and that $R$ and $S$
are type $I$ or type $II$ plane regions. Then 
\begin{eqnarray*}
\iint_{R}f(x,y)dxdy &=&\iint_{S}f(x(u,y),y(u,y))\left| \dfrac{\partial (x,y)%
}{\partial (u,v)}\right| dudv \\
&&
\end{eqnarray*}

\subsection{Related Scientific Notebook Topics}

{\small \hyperref{Calculus, change variable}{}{}{../../Help/DM7-5.tex#Change
of variables}}

{\small \hyperref{Jacobian}{}{}{../../Help/DM9-5.tex#Jacobian}}

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