Circulation is the line integral of a vector field along a closed, directed curve; it measures the field's accumulated tangential component around that curve.

What does curl represent intuitively?

Curl describes the local rotational tendency of a vector field. Its component along a surface normal measures rotation through that surface.

State Stokes' theorem.

For a suitably smooth oriented surface $\Sigma$, the circulation around its boundary equals the curl flux through it: $\oint_{\partial\Sigma}\mathbf{F}\cdot d\mathbf{r}=\iint_{\Sigma}(\nabla\times\mathbf{F})\cdot\mathbf{n}\,dS$.

What is the relationship between the boundary and surface in Stokes' theorem?

The boundary curve $\partial\Sigma$ is the edge of the surface $\Sigma$; the theorem relates a line integral on that edge to a surface integral over the surface.

What does changing curve direction do to circulation?

Reversing the curve direction reverses the sign of the line integral and therefore changes the sign of the circulation.

CalculusLesson 5 of 5 in the course Vector calculus

Stokes' Theorem: Orientation, Proof, and Applications

5 flashcards · 5 quiz questions · 5 match pairs · 5 fill-in sentences · English

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Summary

This lesson introduces Stokes' theorem through the ideas of circulation and local rotation. It explains why a line integral around a boundary can be related to a surface integral over the area inside it.

For a vector field $\mathbf{F}$ and a directed curve $C$ , the line integral $\int_C \mathbf{F}\cdot d\mathbf{r}$ measures the field's accumulated component in the direction of travel. When $C$ is closed, this integral is called the circulation of $\mathbf{F}$ around $C$ .

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What is circulation?: Circulation is the line integral of a vector field along a closed, directed curve; it measures the field's accumulated tangential component around that curve.
What does curl represent intuitively?: Curl describes the local rotational tendency of a vector field. Its component along a surface normal measures rotation through that surface.
State Stokes' theorem.: For a suitably smooth oriented surface $\Sigma$ , the circulation around its boundary equals the curl flux through it: $\oint_{\partial\Sigma}\mathbf{F}\cdot d\mathbf{r}=\iint_{\Sigma}(\nabla\times\mathbf{F})\cdot\mathbf{n}\,dS$ .
What is the relationship between the boundary and surface in Stokes' theorem?: The boundary curve $\partial\Sigma$ is the edge of the surface $\Sigma$ ; the theorem relates a line integral on that edge to a surface integral over the surface.
What does changing curve direction do to circulation?: Reversing the curve direction reverses the sign of the line integral and therefore changes the sign of the circulation.

Quiz questions

For an oriented surface, how is the positive direction around its boundary determined?Answer: By the right-hand rule applied to the chosen normal
If a surface is parametrized by $\mathbf{r}(u,v)$ , what is its oriented vector area element?Answer: $(\mathbf{r}_u\times\mathbf{r}_v)\,du\,dv$
When may a different spanning surface be used to compute curl flux with Stokes' theorem?Answer: When it has the same boundary and a consistent induced orientation
Under what domain condition does a smooth zero-curl field imply a conservative field?Answer: The domain is open and simply connected
In Maxwell–Faraday law, what does Stokes' theorem connect?Answer: The local curl equation for the electric field and its circulation around a loop

Based on the Wikipedia article Stokes' theorem, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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