Change of variables

Sometimes an integral over some region is easier to express or solve in one set of variables than another. For example, an integral over a circle may be easier to do in polar coordinates than in cartesian. Change of variables allows us to express an integral in whatever variables we find most convenient.

First consider change of variables with just a single variable (usually called $u$-substitution). We have some function $u(x): [a,b] \to [c,d]$ that maps one interval to another. If we have an integral of $f(x)$ over $[a,b]$ we can change the integration variable to $u=u(x)$ with the formula:

Since the interval $[c,d]$ might be a different length than $[a,b]$, we need a length scaling factor $du/dx$. When we do a change of variables in several variables, we need to account for the area scaling factor or volume scaling factor the same way.

Remember that the absolute value of the determinant $\left| \mathrm{det} \begin{pmatrix}\mathbf f_1 | \mathbf f_2 | \cdots | \mathbf f_n \end{pmatrix} \right|$ is the volume in $\R^n$ spanned by the vectors $\mathbf f_1$ through $\mathbf f_n$. For some change of variables from $x_1, x_2, \ldots, x_n$ to $u_1, u_2, \ldots, u_n$ the area scaling factor would be $\left| \mathrm{det} \begin{pmatrix}u_1' | u_2' | \cdots | u_n' \end{pmatrix} \right|$. This can also be written as the Jacobian determinant $\left| \frac{\partial(x_1,x_2,\ldots,x_n)}{\partial(u_1,u_2,\ldots,u_n)} \right|$

Steps

The general steps to do a change the variables of $\iint_R f(x,y) dx dy$ from $x,y$ to $u,v$ are:

1. Transform $x=x(u,v)$ and $y=y(u,v)$.

2. Find equation(s) for the boundary curves of $R$ in $x,y$.

3. Rewrite these curves in terms of $u,v$, find the corresponding region $S$.

4. Compute the Jacobian determinant $\left| \frac{\partial(x,y)}{\partial(u,v)} \right|$

5. Substitute $x=x(u,v)$, $y=y(u,v)$, $dxdy=\left| \frac{\partial(x,y)}{\partial(u,v)} \right|dudv$ to get $\iint_R f(x,y) dx dy = \iint_S f(x(u, v), y(u, v)) \left| \frac{\partial(x,y)}{\partial(u,v)} \right|dudv$.

Common changes of variables

To avoid having to recompute the Jacobian determinant, here are some common changes of variables.

Polar coordinates

In polar coordinates $r$ is usually the distance from the origin and $\theta$ is the counter-clockwise angle from the $+x$ axis. In this case, the change of variables is:

• $dA=dxdy=rdrd\theta.$

• $x=r\cos\theta.$

• $y=r\sin\theta.$

Cylindrical coordinates

Cylindrical coordinates are the same as polar coordinates except there is also a $k$ or $z$ component which is just the distance from $x,y$ plane in the $+z$ direction. The change of variables is:

• $dV=dx\,dy\,dz=r\,dr\,d\theta\,dk.$

• $x=r\cos\theta.$

• $y=r\sin\theta.$

• $z=k.$

Spherical coordinates

In spherical coordinates $\rho$ is the distance from the origin, $\phi$ is the angle from the $+z$-axis, and $\theta$ is the same as in cylindrical and polar coordinates.

First, converting from cylindrical coordinates:

• $r=\rho \sin \phi.$

• $\theta=\theta.$

• $k=\rho \cos \phi.$

And from cartesian coordinates:

• $x=\rho\sin\phi\cos\theta.$

• $y=\rho\sin\phi\sin\theta.$

• $z=\rho\cos\phi.$

In general, the volume element $dV = dx\,dy\,dz=r\,dr\,d\theta\,dk = r^2 \sin\phi\, d\rho\, d\phi\, d\theta.$