# Problem-Solving: Multivariable Calculus

Physics I · Multivariable Calculus · https://tryals.app/learn/physics-i/problem-solving-multivariable-calculus

Multivariable calculus requires matching the right technique to the structure of the problem. **Exploiting symmetry** first can instantly simplify or zero out complex integrals.

| Task | Tool |
| --- | --- |
| Rate of change | **Directional derivative** $D_{\hat{u}}f$ |
| Find extrema | $\nabla f = 0$ + second derivative test |
| Constrained opt. | **Lagrange multipliers** |
| Integrate region | Double or triple integral |
| Symmetry/Circles | Polar or spherical coordinates |
| Conservative field | **Potential function** $\phi(B) - \phi(A)$ |
| Closed-curve line | **Green's theorem** |

**Problem checklist:** Sketch the region, identify symmetry, set up correct limits, and verify dimensions.

## Pitfalls & Coordinate Choice

The decisive move happens before integration: choosing coordinates that fit the region. Fighting a sphere with Cartesian limits is a self-inflicted wound.

| Common Mistake | Consequence |
| --- | --- |
| **Wrong limits** | Incorrect iterated integral order |
| **Missing Jacobian** | Forgetting $r$, $r \theta$, or $\rho^{2}\sin\phi$ |
| **Non-closed curves** | Invalid use of Green's theorem |

> **Pitfall:** Never apply Green's theorem to a curve that is not closed. Always check domain boundaries before proceeding.

## Practice questions

6 of this lesson's 13 practice questions, with answers. The full set is in the app.

### 1. Order this complete solution of a classic: find the mass of a hemisphere of radius $R$ whose density grows with height, $\rho = kz$.

**Answer:**

1. Recognize the symmetry and choose spherical coordinates
2. Express the density: $z = \rho\cos\varphi$, so density $= k\rho\cos\varphi$
3. Assemble the integrand with the Jacobian: $k\rho\cos\varphi \cdot \rho^{2}\sin\varphi$
4. Set hemisphere limits: $\rho \in [0,R]$, $\varphi \in [0, \pi/2]$, $\theta \in [0, 2\pi]$
5. Separate and evaluate the three one-variable integrals

**Why:** This is the full multivariable pipeline in one problem: symmetry chooses coordinates, the Jacobian corrects volumes, limits encode the shape, and separability turns a 3D integral into three easy 1D ones (answer: $k\pi R^{4}/4$). Mastering the pipeline matters more than any single integral.

Page: https://tryals.app/practice/physics-i/problem-solving-multivariable-calculus/order-this-complete-solution-of-a-classic-find-the-mass-of-a

### 2. Multivariable problems are often solved by reducing them to single-variable ones. What makes that reduction legitimate?

A. That the operations fix everything but one variable
B. That multivariable functions are sums of single-variable functions
C. That the reduction is an approximation valid near a point
D. That the extra variables can be neglected when they are small

**Answer:** A. That the operations fix everything but one variable

**Why:** The whole toolkit works by holding everything but one thing fixed. That is why single-variable calculus is the prerequisite rather than a special case.

Page: https://tryals.app/practice/physics-i/problem-solving-multivariable-calculus/multivariable-problems-are-often-solved-by-reducing-them-to

### 3. Fubini's theorem allows changing the order of integration in a double integral when the integrand is continuous.

**Answer:** True

**Why:** Continuity on a closed bounded region is sufficient for Fubini's theorem.

Page: https://tryals.app/practice/physics-i/problem-solving-multivariable-calculus/fubinis-theorem-allows-changing-the-order-of-integration-in-a-double

### 4. Match each problem type to the best multivariable technique.

**Answer:**

- Volume under a surface over a disk → Double integral in polar coordinates
- Optimize f on a circle constraint → Lagrange multipliers
- Flux through a closed surface → Divergence theorem
- Rate of change along a curve → Chain rule / directional derivative

**Why:** Choosing the right technique is the key skill in problem-solving.

Page: https://tryals.app/practice/physics-i/problem-solving-multivariable-calculus/match-each-problem-type-to-the-best-multivariable-technique

### 5. Which are valid strategies for evaluating a difficult double integral?

A. Reverse the order of integration
B. Change to polar/cylindrical/spherical coordinates
C. Always use Cartesian coordinates
D. Use symmetry to simplify

**Answer:** A. Reverse the order of integration; B. Change to polar/cylindrical/spherical coordinates; D. Use symmetry to simplify

**Why:** Coordinate changes, order reversal, and symmetry are all powerful techniques.

Page: https://tryals.app/practice/physics-i/problem-solving-multivariable-calculus/which-are-valid-strategies-for-evaluating-a-difficult-double-integral

### 6. Sort each theorem: connects which types of integrals?

**Answer:**

- Line ↔ Double integral: Green's theorem, $\oint P\,dx + Q\,dy = \iint (Q_{x} - P_{y})\,dA$, Area via line integral: $\frac{1}{2}$∮(xdy−ydx)
- Surface ↔ Volume integral: Divergence theorem, ∬F$\cdot dS$ = ∭$\nabla\cdot F$ dV, Gauss's law in electrostatics

**Why:** Green's theorem is 2D (line ↔ area); Divergence theorem is 3D (surface ↔ volume).

Page: https://tryals.app/practice/physics-i/problem-solving-multivariable-calculus/sort-each-theorem-connects-which-types-of-integrals