16.2 Change of variables theorem for double and triple integrals

Change of Variables Theorem

Applying the Change of Variables Theorem

The change of variables theorem lets you evaluate a double or triple integral by switching to a new coordinate system where the integrand, the region, or both become simpler. You replace the original variables with new ones and multiply by the absolute value of the Jacobian determinant to correctly scale the area or volume elements.

For double integrals, you substitute new variables $(u, v)$ for $(x, y)$. The theorem states:

$\iint_{R} f(x,y) \, dA = \iint_{S} f\!\bigl(x(u,v),\, y(u,v)\bigr) \left| \frac{\partial(x,y)}{\partial(u,v)} \right| du \, dv$

where $R$ is the original region, $S$ is the corresponding region in the $(u,v)$-plane, and the absolute value of the Jacobian determinant accounts for how area elements stretch or compress under the transformation.

For triple integrals, you substitute $(u, v, w)$ for $(x, y, z)$:

$\iiint_{R} f(x,y,z) \, dV = \iiint_{S} f\!\bigl(x(u,v,w),\, y(u,v,w),\, z(u,v,w)\bigr) \left| \frac{\partial(x,y,z)}{\partial(u,v,w)} \right| du \, dv \, dw$

The logic is the same: the Jacobian determinant scales the volume element from $(x,y,z)$-space to $(u,v,w)$-space.

Choosing good new variables is the whole art of this technique. You want variables that:

• Simplify the integrand (e.g., turning $x^2/a^2 + y^2/b^2$ into $r^2$)
• Transform the region into a simpler shape (rectangle, disk, box)
• Make the integration limits constant or easy to determine

For example, a double integral over an elliptical region $x^2/a^2 + y^2/b^2 \leq 1$ becomes an integral over a unit disk when you set $u = x/a$ and $v = y/b$, and then you can switch to polar coordinates on that disk.

Steps for Applying the Theorem

1. Define the transformation. Write $x, y$ (or $x, y, z$) as functions of the new variables $(u, v)$ or $(u, v, w)$.
2. Compute the Jacobian determinant. Build the matrix of partial derivatives and take its determinant (details below).
3. Transform the region. Express the boundaries of the original region $R$ in terms of the new variables to get the new region $S$.
4. Rewrite the integrand. Substitute the transformation equations into $f$ so everything is in the new variables.
5. Set up and evaluate. Write the new integral over $S$ with the absolute value of the Jacobian as the scaling factor, then integrate.

Transformation and Jacobian

Transforming Regions and Integration Limits

When you change variables, the shape of the integration region typically changes. A rectangular region in Cartesian coordinates might become a polar rectangle; an ellipse might become a circle or a rectangle.

To find the new limits of integration:

• Take each boundary curve (or surface, in 3D) of the original region and substitute the transformation equations.
• Solve for the corresponding equations in the new variables.
• Use those equations to determine the limits on each new variable.

For example, a circular disk $x^2 + y^2 \leq a^2$ in Cartesian coordinates maps to $0 \leq r \leq a$, $0 \leq \theta \leq 2\pi$ in polar coordinates. The region became a rectangle in the $(r, \theta)$-plane, which is much easier to integrate over.

Jacobian Determinant and Its Role

The Jacobian determinant is the scaling factor that tells you how much the transformation stretches or compresses area (2D) or volume (3D) at each point. Without it, your integral would give the wrong answer because the "size" of each infinitesimal patch changes under the mapping.

For double integrals, the Jacobian matrix and its determinant are:

$\frac{\partial(x,y)}{\partial(u,v)} = \det \begin{pmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \$4pt] \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{pmatrix} = \frac{\partial x}{\partial u}\frac{\partial y}{\partial v} - \frac{\partial x}{\partial v}\frac{\partial y}{\partial u}$

You then take the absolute value of this determinant in the integral.

For triple integrals, the Jacobian is the determinant of the $3 \times 3$ matrix of partial derivatives:

$\frac{\partial(x,y,z)}{\partial(u,v,w)} = \det \begin{pmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \$4pt] \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \$4pt] \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w} \end{pmatrix}$

Common Jacobians Worth Knowing

These come up constantly, so it's worth memorizing them:

• Polar coordinates ($x = r\cos\theta,\; y = r\sin\theta$): Jacobian is $r$, so $dA = r \, dr \, d\theta$
• Cylindrical coordinates ($x = r\cos\theta,\; y = r\sin\theta,\; z = z$): Jacobian is $r$, so $dV = r \, dr \, d\theta \, dz$
• Spherical coordinates ($x = \rho\sin\phi\cos\theta,\; y = \rho\sin\phi\sin\theta,\; z = \rho\cos\phi$): Jacobian is $\rho^2 \sin\phi$, so $dV = \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta$

Notice that the Jacobian for polar coordinates is just $r$, not a complicated expression. That $r$ is exactly why $dA = r\,dr\,d\theta$ in polar coordinates: it comes directly from the change of variables theorem. The same reasoning produces the $\rho^2\sin\phi$ factor in spherical coordinates. Every "special" area or volume element you've seen in earlier units is really just the Jacobian at work.