Change of Variables in Multiple Integrals

Introduction

When implementing a change in variables in an integral, in addition to the bounds and the function to integrate, the infinitesimal needs to be changed as well.

The Essentials

To change \( dA \) or \( dV \) into new variables, the Jacobian is found. The Jacobian is the determinant of the partials of \( x \), \( y \), \( z \). with respect to the new variables, usually \( u \), \( v \), \( w \). In two dimensions:

\[ J(u,v)=\frac{|\partial(x,y)|}{|\partial(u,v)|}=\begin{vmatrix} \\ \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \\ \end{vmatrix}=\left(\frac{\partial x}{\partial u}\frac{\partial y}{\partial v}-\frac{\partial x}{\partial v}\frac{\partial y}{\partial u}\right) \\ \]

In three dimensions:

\[ J(u,v,w)=\frac{|\partial(x,y,z)|}{|\partial(u,v,w)|}=\begin{vmatrix} \\ \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w} \\ \end{vmatrix} \]

So that \( dA=Jdudv \) and \( dV=Jdudvdw \). To make a change of variables in two dimensions:

\[ \iint_R f(x,y)dA=\iint_S f(g(u,v),h(u,v)) Jdudv \]

With three variables:

\[ \iiint_R f(x,y)dA=\iiint_G f(g(u,v,w),h(u,v,w),k(u,v,w)) Jdudvdw \]

Example

Derive the formula for polar coordinates.

In polar coordinates, \( f(x, y) \) becomes \( f(g(r,\theta), h(r, \theta)) \) where \( x=g(r,\theta)=r\cos(\theta) \) and \( y=h(r,\theta)=r\sin(\theta) \). The differential \( dA \) will become \( Jdrd\theta \). To find J:

\[ J=\begin{vmatrix} \\ x_r & x_\theta \\ y_r & y_\theta \\ \end{vmatrix}=\begin{vmatrix} \\ \cos(\theta) & -r\sin(\theta) \\ \sin(\theta) & r\cos(\theta) \\ \end{vmatrix}=r\cos^2(\theta)-(-r\sin^2(\theta))=r \\ \]

Therefore to convert rectangular coordinates to polar coordinates:

\[ \iint_{R}f(x,y)dA=\iint_S f(r\cos(\theta),r\sin(\theta))rdrd \\ \theta \]

Practice

1) Derive the formula for spherical coordinates

2) Derive the formula for cylindrical coordinates

Solutions:

1) \( dV=\rho^2\sin(\phi)d\rho d\theta d\phi \)

2) \( dV=rdrd\theta dz \)

More Resources

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