# Change of Variables and the Jacobian

Physics I · Multivariable Calculus · https://tryals.app/learn/physics-i/change-of-variables-and-the-jacobian

## Change of Variables & The Jacobian

A **change of variables** transforms a hard integral into a simpler coordinate system, generalising single-variable substitution $dx = g'(u)du$ to multiple dimensions.

If transformation $T: (u,v) \mapsto (x,y)$ maps region $S$ to $R$, the integral becomes:

$$\iint_R f(x,y)\,dA = \iint_S f(x(u,v),y(u,v))\;|J|\,du\,dv$$

Here, $J$ is the **Jacobian determinant**, which measures how the transformation locally **scales area** or volume.

$$J = \frac{\partial(x,y)}{\partial(u,v)} = \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{vmatrix}$$

## Standard Jacobians & Pitfalls

Certain coordinate systems appear frequently. Memorise their scaling factors:

| Coordinate System | Variables | Scaling Factor $|J|$ |
| --- | --- | --- |
| **Polar** | $(r,\theta)$ | $r$ |
| **Cylindrical** | $(r,\theta,z)$ | $r$ |
| **Spherical** | $(\rho,\phi,\theta)$ | $\rho^{2}\sin\phi$ |

> **Common pitfall:** Forgetting the absolute value $|J|$ or omitting the scaling factor entirely. If a disk integral returns $2\pi R$ instead of $\pi R^2$, a missing polar $r$ is always the culprit.

## Practice questions

6 of this lesson's 14 practice questions, with answers. The full set is in the app.

### 1. Sort each statement as True or False about the Jacobian.

**Answer:**

- True: The Jacobian is a determinant of partial derivatives, |J| scales area/volume elements under coordinate change, A Jacobian of 0 means the transformation collapses dimension, For Cartesian-to-Cartesian, $J = 1$ (identity)
- False: The Jacobian for polar coords is $r^{2}$, The Jacobian is always positive

**Why:** The Jacobian can be negative (orientation-reversing); we take |J| for integrals.

Page: https://tryals.app/practice/physics-i/change-of-variables-and-the-jacobian/sort-each-statement-as-true-or-false-about-the-jacobian

### 2. When you switch a double integral to polar coordinates, $dA$ becomes $r\,dr\,d\theta$, not just $dr\,d\theta$. What does that extra $r$ pay for?

A. It converts degrees to radians
B. It is a convention with no geometric meaning
C. It corrects for the origin being excluded
D. Area cells far from the origin are bigger

**Answer:** D. Area cells far from the origin are bigger

**Why:** The polar grid’s cells grow with distance: width $r\,d\theta$ times depth $dr$. The factor $r$, the Jacobian of the transformation, is the local area-conversion rate. Forgetting it is the most common change-of-variables error in the subject.

Page: https://tryals.app/practice/physics-i/change-of-variables-and-the-jacobian/when-you-switch-a-double-integral-to-polar-coordinates-da-becomes-r

### 3. The Jacobian appears as a multiplying factor in every change of variables. What would go wrong if it were omitted?

A. Nothing, provided the transformation is linear
B. The integral would count volume in the wrong units
C. The limits of integration would be incorrect
D. The integral would have the right value but the wrong sign

**Answer:** B. The integral would count volume in the wrong units

**Why:** The Jacobian converts $du\,dv$ into real area. Omitting it weights every region equally when the map does not, which is why $r$ appears in polar and $\rho^{2}\sin\varphi$ in spherical.

Page: https://tryals.app/practice/physics-i/change-of-variables-and-the-jacobian/the-jacobian-appears-as-a-multiplying-factor-in-every-change-of

### 4. Order the steps of a change of variables in a double integral.

**Answer:**

1. Define new variables $u = u(x,y), v = v(x,y)$
2. Solve for $x(u,v)$ and $y(u,v)$
3. Compute the Jacobian $\partial(x,y)/\partial(u,v)$
4. Transform the integrand and limits
5. Evaluate ∬ f($x(u,v)$,$y(u,v)$) |J| du dv

**Why:** Change of variables: substitute, scale by |J|, adjust limits.

Page: https://tryals.app/practice/physics-i/change-of-variables-and-the-jacobian/order-the-steps-of-a-change-of-variables-in-a-double-integral

### 5. Match each coordinate change to its volume/area factor (its Jacobian).

**Answer:**

- Polar $(r, \theta)$ → $r$
- Cylindrical $(r, \theta, z)$ → $r$ (the $z$ direction adds no distortion)
- Spherical $(\rho, \varphi, \theta)$ → $\rho^{2}\sin\varphi$
- Linear map $u = ax + by,\ v = cx + dy$ → $1/|ad - bc|$ (from $dx\,dy$ to $du\,dv$)

**Why:** Jacobians are exchange rates between coordinate area and true area. Cylindrical inherits polar’s $r$ untouched; spherical pays $\rho^{2}\sin\varphi$ (shells grow with radius, and longitude lines crowd at the poles); linear maps pay a constant determinant.

Page: https://tryals.app/practice/physics-i/change-of-variables-and-the-jacobian/match-each-coordinate-change-to-its-volume-area-factor-its-jacobian

### 6. Using polar coordinates, $\iint r\,dr\,d\theta$ over the full disk of radius 2 equals $2\pi \times \int_0^2 r\,dr$. Evaluate the area of the disk. **Set the slider to your answer.**

**Answer:** 12.566 (within ±0.4)

**Why:** $2\pi \times 2 = 4\pi \approx 12.57$, reproducing $\pi R^{2}$ from first principles. The Jacobian $r$ is exactly what makes the polar computation land on the true area.

Page: https://tryals.app/practice/physics-i/change-of-variables-and-the-jacobian/using-polar-coordinates-r-dr-d-over-the-full-disk-of-radius-2