Change of Variables and the Jacobian Cheat Sheet

Change of variables rewrites an integral using new variables that make the region, integrand, or symmetry easier to handle. This cheat sheet focuses on the Jacobian, the scale factor that corrects area or volume after a transformation. College calculus students need it for double integrals, triple integrals, polar coordinates, cylindrical coordinates, spherical coordinates, and substitutions over non-rectangular regions.

It is especially useful when a region becomes simpler in variables such as $u$ , $v$ , $r$ , $\theta$ , $\rho$ , or $\phi$ .

The central idea is that a transformation such as $x = x(u,v)$ and $y = y(u,v)$ changes small area elements by the factor $\left|\frac{\partial(x,y)}{\partial(u,v)}\right|$ . In three dimensions, $dV$ changes by $\left|\frac{\partial(x,y,z)}{\partial(u,v,w)}\right|\,du\,dv\,dw$ . Standard coordinate systems have built-in Jacobians, including $dA = r\,dr\,d\theta$ for polar coordinates and $dV = \rho^2\sin\phi\,d\rho\,d\phi\,d\theta$ for spherical coordinates.

Always transform both the integrand and the bounds before integrating.

Key Facts

For a two-variable transformation $x = x(u,v)$ and $y = y(u,v)$ , the area element becomes $dA = \left|\frac{\partial(x,y)}{\partial(u,v)}\right|\,du\,dv$ .
The two-variable Jacobian determinant is $\frac{\partial(x,y)}{\partial(u,v)} = \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{vmatrix} = \frac{\partial x}{\partial u}\frac{\partial y}{\partial v} - \frac{\partial x}{\partial v}\frac{\partial y}{\partial u}$ .
For a three-variable transformation $x = x(u,v,w)$ , $y = y(u,v,w)$ , and $z = z(u,v,w)$ , the volume element becomes $dV = \left|\frac{\partial(x,y,z)}{\partial(u,v,w)}\right|\,du\,dv\,dw$ .
A double integral changes variables by $\iint_R f(x,y)\,dA = \iint_S f(x(u,v),y(u,v))\left|\frac{\partial(x,y)}{\partial(u,v)}\right|\,du\,dv$ .
Polar coordinates use $x = r\cos\theta$ , $y = r\sin\theta$ , and $dA = r\,dr\,d\theta$ .
Cylindrical coordinates use $x = r\cos\theta$ , $y = r\sin\theta$ , $z = z$ , and $dV = r\,dr\,d\theta\,dz$ .
Spherical coordinates use $x = \rho\sin\phi\cos\theta$ , $y = \rho\sin\phi\sin\theta$ , $z = \rho\cos\phi$ , and $dV = \rho^2\sin\phi\,d\rho\,d\phi\,d\theta$ .
If a transformation has inverse Jacobian $\frac{\partial(u,v)}{\partial(x,y)}$ , then $\left|\frac{\partial(x,y)}{\partial(u,v)}\right| = \frac{1}{\left|\frac{\partial(u,v)}{\partial(x,y)}\right|}$ when the inverse exists.

Vocabulary

Change of Variables: A method for rewriting an integral in new variables so the region or integrand becomes easier to evaluate.
Jacobian: The determinant that measures how a transformation locally scales area or volume.
Transformation: A rule such as $x = x(u,v)$ and $y = y(u,v)$ that maps points from one coordinate system to another.
Area Element: The small piece of area in an integral, such as $dA$ or its transformed form $\left|J\right|\,du\,dv$ .
Volume Element: The small piece of volume in an integral, such as $dV$ or its transformed form $\left|J\right|\,du\,dv\,dw$ .
One-to-One Transformation: A transformation that maps each point in the new variable region to exactly one point in the original region.

Common Mistakes to Avoid

Forgetting the absolute value of the Jacobian is wrong because area and volume scale factors must be nonnegative, so use $\left|J\right|$ instead of $J$ .
Changing only the bounds but not the integrand is wrong because every occurrence of $x$ , $y$ , and $z$ must be replaced by the new variable formulas.
Using $dA = dr\,d\theta$ in polar coordinates is wrong because the correct area element is $dA = r\,dr\,d\theta$ .
Mixing up spherical angle conventions is wrong because in standard calculus notation $\phi$ is measured from the positive $z$ -axis and $dV = \rho^2\sin\phi\,d\rho\,d\phi\,d\theta$ .
Assuming any transformation is valid everywhere is wrong because the Jacobian can be zero or the mapping can fail to be one-to-one on the chosen region.

Practice Questions

1 Use polar coordinates to evaluate $\iint_R (x^2 + y^2)\,dA$ where $R$ is the disk $x^2 + y^2 \le 9$ .
2 For $x = u^2 - v^2$ and $y = 2uv$ , compute $\frac{\partial(x,y)}{\partial(u,v)}$ .
3 Use spherical coordinates to set up, but not evaluate, $\iiint_B z^2\,dV$ where $B$ is the ball $x^2 + y^2 + z^2 \le 16$ .
4 Explain why the factor $\left|J\right|$ is needed when changing variables, even if the transformation reverses orientation.

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