Chapter 33 — Exercises

36 problems on the Jacobian determinant and change of variables. Tiered ⭐ (mechanical) to ⭐⭐⭐⭐ (synthesis and proof). Work each by hand; the SymPy from Section 33.11 is for checking, never for doing. Answers to selected problems appear in appendices/answers-to-selected.md.

How to Use These Exercises

The chapter gave you one formula and one idea: a smooth map looks locally like its Jacobian matrix, and the absolute value of that matrix's determinant is the local area- or volume-scaling factor (Section 33.3, Section 33.6). Everything below is practice in three skills:

1. Computing $\det J_T$ from partial derivatives (Section 33.3).
2. Applying the change-of-variables formula to a double or triple integral (Section 33.4, Section 33.7).
3. Designing a transformation that simplifies a region or an integrand (the Worked Examples of Section 33.4).

Always state the direction of your transformation explicitly. As Section 33.4's pitfall warns, $\det \partial(x,y)/\partial(u,v)$ and $\det \partial(u,v)/\partial(x,y)$ are reciprocals (Section 33.8), and the Jacobian factor always multiplies the new differentials.

Part A — Computing Jacobian Determinants (⭐)

A1. ⭐ Compute the Jacobian determinant of the linear map $x = 3u - v$, $y = u + 2v$. Is area preserved, expanded, or shrunk?

A2. ⭐ For $x = u^2 - v^2$, $y = 2uv$, compute $\det \dfrac{\partial(x,y)}{\partial(u,v)}$ and simplify. (This is the complex squaring map $z \mapsto z^2$.)

A3. ⭐ Verify by direct computation that the polar map $x = r\cos\theta$, $y = r\sin\theta$ has $\det J = r$ (Section 33.3).

A4. ⭐ Compute $\det J$ for the axis stretch $x = 5u$, $y = 2v$. What is the area of the image of the unit square $[0,1]\times[0,1]$?

A5. ⭐ For the rotation $x = u\cos\alpha - v\sin\alpha$, $y = u\sin\alpha + v\cos\alpha$ (fixed angle $\alpha$), show $\det J = 1$. Why is this geometrically obvious?

A6. ⭐ Compute the Jacobian determinant of $x = u + v$, $y = u - v$, $z = w$ (a 3D map). Reduce the $3\times3$ determinant to a $2\times2$ by expanding along the bottom row, as in the cylindrical derivation (Section 33.7).

Part B — Change of Variables in Double Integrals (⭐⭐)

B1. ⭐⭐ Use $u = x + y$, $v = x - y$ to evaluate $\displaystyle\iint_R (x+y)\,dA$ where $R$ is the square with vertices $(0,0)$, $(1,1)$, $(2,0)$, $(1,-1)$. (Find the image region $S$ first.)

B2. ⭐⭐ Evaluate $\displaystyle\iint_R e^{(x-y)/(x+y)}\,dA$ over the triangle $R$ bounded by $x = 0$, $y = 0$, $x + y = 1$, using $u = x + y$, $v = x - y$.

B3. ⭐⭐ Use the stretch $x = 2u$, $y = 3v$ to evaluate the area $\displaystyle\iint_E \,dA$ over the ellipse $\dfrac{x^2}{4} + \dfrac{y^2}{9} \le 1$. Confirm you recover $\pi a b$ (Section 33.5).

B4. ⭐⭐ Evaluate $\displaystyle\iint_{\mathbb{R}^2} e^{-(x^2+y^2)}\,dA$ in polar coordinates and confirm the answer is $\pi$ (Worked Example 4, Section 33.4). Then deduce $\int_{-\infty}^\infty e^{-x^2}\,dx = \sqrt{\pi}$.

B5. ⭐⭐ The parallelogram $R$ is bounded by $y = 2x$, $y = 2x - 4$, $y = x$, $y = x + 1$. Choose new variables $u, v$ from the boundary lines, find $\det J$, and compute the area of $R$.

B6. ⭐⭐ Evaluate $\displaystyle\iint_R \cos\!\left(\frac{x-y}{x+y}\right) dA$ over the trapezoid with vertices $(1,0)$, $(2,0)$, $(0,2)$, $(0,1)$ using $u = x + y$, $v = x - y$.

B7. ⭐⭐ Using $u = xy$, $v = y/x$, set up (do not fully evaluate) $\displaystyle\iint_R f\,dA$ over the first-quadrant region bounded by $xy = 1$, $xy = 3$, $y = x$, $y = 4x$. Find $|\det J|$ in terms of $u, v$ by computing $\det \partial(u,v)/\partial(x,y)$ and inverting (Section 33.8).

Part C — Deriving the Coordinate Jacobians (⭐⭐)

C1. ⭐⭐ Derive the polar Jacobian $|\det J| = r$ from scratch and explain in one sentence why the factor grows with $r$ (Section 33.3).

C2. ⭐⭐ Derive the cylindrical volume element $dV = r\,dr\,d\theta\,dz$ by expanding the $3\times3$ Jacobian along its bottom row (Section 33.7).

C3. ⭐⭐⭐ Derive the spherical Jacobian $|\det J| = \rho^2\sin\phi$ by hand for $x = \rho\sin\phi\cos\theta$, $y = \rho\sin\phi\sin\theta$, $z = \rho\cos\phi$ (Section 33.7). Show every step of the cofactor expansion.

C4. ⭐⭐ Without computing a determinant, explain why the spherical volume element vanishes at $\phi = 0$ and at $\rho = 0$ (Section 33.7's Check Your Understanding).

C5. ⭐⭐ A "parabolic coordinate" map is $x = \tfrac12(u^2 - v^2)$, $y = uv$. Compute $\det J$ and identify the curves $u = \text{const}$ and $v = \text{const}$ as parabolas.

Part D — Triple Integrals and 3D Change of Variables (⭐⭐⭐)

D1. ⭐⭐⭐ Use spherical coordinates to find the volume of the ball $x^2 + y^2 + z^2 \le a^2$. Confirm $\frac{4}{3}\pi a^3$.

D2. ⭐⭐⭐ Evaluate $\displaystyle\iiint_B (x^2 + y^2 + z^2)\,dV$ over the unit ball using spherical coordinates, carrying the $\rho^2\sin\phi$ factor.

D3. ⭐⭐⭐ Find the mass of a solid cylinder $x^2 + y^2 \le R^2$, $0 \le z \le h$, with density $\delta(x,y,z) = k(x^2 + y^2)$, using cylindrical coordinates (Section 33.7).

D4. ⭐⭐⭐ Use the linear map $x = u$, $y = u + v$, $z = u + v + w$ to evaluate $\displaystyle\iiint_R \,dV$ where $R$ is the image of the unit cube $[0,1]^3$. Compute $\det J$ and the volume of $R$.

D5. ⭐⭐⭐ A solid is the image of the unit ball under the stretch $x = au$, $y = bv$, $z = cw$ (an ellipsoid). Use $\det J = abc$ to show its volume is $\frac{4}{3}\pi abc$.

Part E — Probability Density Transformations (⭐⭐⭐)

E1. ⭐⭐⭐ Let $X$ be uniform on $[0,1]$ and define $Y = -\ln X$. Use the one-dimensional density rule $p_Y(y) = p_X(x)\,\lvert dx/dy\rvert$ to show $Y$ is exponentially distributed: $p_Y(y) = e^{-y}$ for $y \ge 0$ (Section 33.9, single-variable case).

E2. ⭐⭐⭐ Let $(X, Y)$ have the standard 2D Gaussian density $p_{XY}(x,y) = \frac{1}{2\pi}e^{-(x^2+y^2)/2}$. Change to polar $(R, \Theta)$ via $x = r\cos\theta$, $y = r\sin\theta$. Using $p_{R\Theta}(r,\theta) = p_{XY}\,\lvert\det J\rvert$ with $\lvert\det J\rvert = r$, find the joint density of $(R, \Theta)$ and show $R$ and $\Theta$ are independent.

E3. ⭐⭐⭐ (Box–Muller, Section 33.9). Let $U_1, U_2$ be independent uniform on $(0,1)$. Define $X = \sqrt{-2\ln U_1}\cos(2\pi U_2)$ and $Y = \sqrt{-2\ln U_1}\sin(2\pi U_2)$. Using $p_{XY} = p_{U_1 U_2}\,\lvert\det J_{T^{-1}}\rvert$, show $(X,Y)$ has the standard 2D Gaussian density. (Hint: the inverse map sends $(x,y)$ to $u_1 = e^{-(x^2+y^2)/2}$, $u_2 = \frac{1}{2\pi}\arctan(y/x)$.)

E4. ⭐⭐⭐ Let $(X,Y)$ be uniform on the unit disk, density $p_{XY} = 1/\pi$ inside. Find the density of $S = X^2 + Y^2$. (Switch to polar; $S = R^2$.)

Part F — Applied (≥ 2 fields) (⭐⭐⭐)

F1. ⭐⭐⭐ (Physics — moment of inertia). Find the moment of inertia $I = \iiint r^2\,\delta\,dV$ (with $r^2 = x^2 + y^2$, the squared distance from the $z$-axis) of a uniform solid sphere of radius $a$, mass $M$, about a diameter. Use spherical coordinates and write $r^2 = \rho^2\sin^2\phi$. Confirm $I = \frac{2}{5}Ma^2$.

F2. ⭐⭐⭐ (Economics — decorrelating an integral). Two correlated quantities have a cost density proportional to $\exp\!\big(-(x^2 - xy + y^2)\big)$ over $\mathbb{R}^2$. The cross term $xy$ is awkward. Find a linear change of variables (the $45°$ rotation $x = (u+v)/\sqrt2$, $y = (u-v)/\sqrt2$) that removes the cross term, compute the constant Jacobian, and evaluate $\iint_{\mathbb{R}^2} \exp\!\big(-(x^2 - xy + y^2)\big)\,dA$.

F3. ⭐⭐⭐ (Engineering — skewed sensor region). A stress integral $\iint_R (x + 2y)\,dA$ must be computed over the parallelogram bounded by $x - 2y = 0$, $x - 2y = 3$, $3x + y = 1$, $3x + y = 5$. Choose $u = x - 2y$, $v = 3x + y$, find $\det \partial(u,v)/\partial(x,y)$, invert to get $|\det J|$, and evaluate.

F4. ⭐⭐⭐ (Biology — spherical organ). A spherical cell of radius $a$ has a chemical concentration $c(\rho) = c_0(1 - \rho/a)$ decreasing linearly from center to surface. Find the total amount $\iiint c\,dV$ inside the cell using spherical coordinates.

Part G — Synthesis and Proof (⭐⭐⭐⭐)

G1. ⭐⭐⭐⭐ Prove that for an invertible, continuously differentiable $T:(u,v)\to(x,y)$, the Jacobian determinants of $T$ and $T^{-1}$ are reciprocals (Section 33.8). Use the chain rule $J_{T^{-1}}J_T = I$ and the multiplicativity of the determinant.

G2. ⭐⭐⭐⭐ Chained transformations. Compute $\displaystyle\iint_E \left(\frac{x^2}{a^2} + \frac{y^2}{b^2}\right) dA$ over the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} \le 1$ by first stretching to the unit disk ($x = au$, $y = bv$), then using polar (Worked Example 2, Section 33.4). Confirm the answer $\frac{\pi a b}{2}$ and explain why the two Jacobian factors ($ab$ and $\rho$) multiply.

G3. ⭐⭐⭐⭐ Designing from a family of curves. The first-quadrant region $R$ is bounded by $y = x^3$, $y = 4x^3$, $xy = 1$, and $xy = 4$. Introduce coordinates labeling the two families ($u = y/x^3$, $v = xy$), find $|\det J|$, and set up $\iint_R \,dA$ as an integral over a rectangle. (Compute $\det\partial(u,v)/\partial(x,y)$ and invert.)

G4. ⭐⭐⭐⭐ (Normalizing-flow flavor, data science). A single "flow" layer in 1D is the invertible map $y = T(x) = x + \tanh(x)$, applied to a base density $p_X$. (a) Show $T$ is strictly increasing, hence invertible. (b) Write the transformed density $p_Y(y) = p_X(T^{-1}(y))\,\lvert dx/dy\rvert$ in terms of $\dfrac{dy}{dx} = 1 + \operatorname{sech}^2 x$. (c) Explain, referencing Section 33.9, why the model is designed so $dy/dx$ (the 1D "Jacobian") is cheap, and what the $n$-dimensional analog (triangular $J$) buys you.

G5. ⭐⭐⭐⭐ Liouville flavor (physics). A map on phase space is $q = Q$, $p = P + f(Q)$ for any smooth $f$ (a "shear"). Show $\det J = 1$, so the map is area-preserving in $(q,p)$. Connect this to Liouville's theorem (Section 33.10): why must any Hamiltonian flow have unit Jacobian?

Tier Summary

Add to Your Modeling Portfolio. Problems F1 (Physics), F2 (Economics), F4 (Biology), and G4 (Data Science) each extend one track of the portfolio described in Section 33.13. Pick the one matching your chosen track and write up the coordinate change as a portfolio entry, explaining why the Jacobian was the tool that let the integral be posed in the natural coordinate system.