Foundations — Stokes' theorem — statement, curl-circulation connection

Before you can read ${\oint }_C\mathbf{F}\cdot d\mathbf{r}={\iint }_S(\nabla \times \mathbf{F})\cdot d\mathbf{S}$, you must be able to read each mark on the page. This note earns every one of them, in an order where each depends only on the ones before it.

0. The stage: ${\mathbb{R}}^3$ and a point

A single point is one such triple. A function of position assigns a number (or an arrow) to every point. That second idea — an arrow at every point — is the star of the show.

1. Vectors and the arrow $\mathbf{F}$

Because the arrow changes from place to place, its components are functions of position: $\mathbf{F}(x,y,z)=(P(x,y,z),Q(x,y,z),R(x,y,z)).$

2. The dot product $\cdot$ — "how much do two arrows agree?"

3. Curves, the boundary $C$, and $\partial S$

Putting these together, ${\oint }_C\mathbf{F}\cdot d\mathbf{r}$ = "walk once around the loop, and at each step add up how much the field pushes you along your step." That total is called the circulation. (Deep dive: Line Integrals of Vector Fields.)

4. Orientation and the right-hand rule

5. The normal $\hat{\mathbf{n}}$ and the flux element $d\mathbf{S}$

6. Partial derivatives ${\partial }_x$ — "steepness in one direction"

The curly $\partial$ (not the straight $d$) is a reminder: "several variables are present; I'm changing just this one."

7. Curl $\nabla \times \mathbf{F}$ — the local spin meter

8. The whole sentence, symbol by symbol

Now every mark is earned. Read Stokes' theorem as English:

$\underset{\text{around the loop}}{\underbrace{{\oint }_C}}\underset{\text{push along each step}}{\underbrace{\mathbf{F}\cdot d\mathbf{r}}}=\underset{\text{over the sheet}}{\underbrace{{\iint }_S}}\underset{\text{local spin}}{\underbrace{(\nabla \times \mathbf{F})}}\cdot \underset{\text{through each patch}}{\underbrace{d\mathbf{S}}}$

Prerequisite map

Equipment checklist

Test yourself — cover the right side of each line.