# Stokes' Theorem

Physics I · Differential Equations and Vector Calculus · https://tryals.app/learn/physics-i/stokes-theorem

## Stokes' Theorem Basics

**Stokes' theorem** links a boundary line integral to an interior surface integral:

$$\oint_C \vec{F}\cdot d\vec{r} = \iint_S (\nabla\times\vec{F})\cdot d\vec{S}$$

Here, $C = \partial S$ is the boundary, oriented by the right-hand rule.

**Physical meaning**: Circulation around a loop equals the total curl piercing any surface it bounds. Boundary behavior dictates interior sums.

| Theorem | Boundary | Interior | Core Concept |
| --- | --- | --- | --- |
| **Fund. Thm.** | Endpoints $a, b$ | Curve $[a, b]$ | Net change |
| **Stokes** | Loop $C$ | Surface $S$ | Circulation |
| **Divergence** | Closed Surface | Solid Volume | Net flux |

## Special Cases & Pitfalls

**Conservative fields**: When $\nabla\times\vec{F} = \vec{0}$, the line integral $\oint \vec{F}\cdot d\vec{r} = 0$.

**Green's theorem** is simply Stokes' theorem specialized to $\mathbb{R}^{2}$:

$$\oint_C (P\,dx + Q\,dy) = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)dA$$

> **Common pitfall**: The curve's direction and the surface's normal vector are strictly chained by the right-hand rule. Choosing them independently yields an answer off by a negative sign.

## Practice questions

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

### 1. Match each integral theorem to its dimensionality.

**Answer:**

- Fundamental theorem of calculus → 1D: $f(b)$−$f(a)$ = $\int f' dx$
- Green's theorem → 2D: line integral ↔ area integral
- Stokes' theorem → 3D: line integral ↔ surface integral
- Divergence theorem → 3D: surface integral ↔ volume integral

**Why:** Each theorem relates a boundary integral to an interior integral, at increasing dimension.

Page: https://tryals.app/practice/physics-i/stokes-theorem/match-each-integral-theorem-to-its-dimensionality

### 2. Two different surfaces, a flat disk and a tall dome, share the **same boundary circle** $C$. How do the fluxes of $\nabla\times\vec{F}$ through them compare?

A. They cannot be compared without knowing $\vec{F}$
B. They are equal: both tie to the same line integral
C. The dome catches more, having more area
D. They agree only if $\vec{F}$ is conservative

**Answer:** B. They are equal: both tie to the same line integral

**Why:** Curl flux is determined entirely by the rim: any surface spanning $C$ — disk, dome, or crumpled sheet — yields the identical value. A soap film can bulge freely on a wire loop without changing the circulation around the wire.

Page: https://tryals.app/practice/physics-i/stokes-theorem/two-different-surfaces-a-flat-disk-and-a-tall-dome-share-the-same

### 3. Match each integral theorem to what it equates.

**Answer:**

- Fundamental theorem for line integrals → Endpoint values ↔ integral of gradient along a curve
- Green’s theorem → Loop circulation in the plane ↔ curl over the enclosed area
- Stokes’ theorem → Loop circulation in 3D ↔ curl flux through a spanning surface
- Divergence theorem → Flux through a closed surface ↔ divergence over the volume

**Why:** These are one theorem climbing a ladder of dimensions, each equates a boundary quantity with an interior derivative. Recognizing which rung a problem sits on tells you which conversion will crack it.

Page: https://tryals.app/practice/physics-i/stokes-theorem/match-each-integral-theorem-to-what-it-equates

### 4. Stokes' theorem generalizes Green's theorem to surfaces in 3D.

**Answer:** True

**Why:** Green's theorem is the special case of Stokes' theorem for flat regions in the xy-plane.

Page: https://tryals.app/practice/physics-i/stokes-theorem/stokes-theorem-generalizes-greens-theorem-to-surfaces-in-3d