Questions: Stokes' Theorem: Circulation and Curl

Question 1

You need to compute ∮_C F·dr where C is a complicated 3D curve. Applying Stokes' theorem, you may:

• A. Replace C with any simpler closed curve in the same plane
• B. Choose any surface spanning C and compute the flux of curl(F) through it
• C. Only use the planar surface bounded directly by C
• D. Replace the line integral with a volume integral over the region enclosed by C

Answer: (B)

Stokes' theorem guarantees ∮_C F·dr = ∬_S (∇×F)·n dS for ANY surface S bounded by C — flat disk, hemisphere, saddle, anything. The freedom to choose S is the theorem's strategic power: pick whichever surface makes the flux integral tractable. Option C (using only the flat surface) is a special case, not a requirement.

Question 2

A vector field F satisfies ∇×F = 0 everywhere. What does Stokes' theorem immediately imply about line integrals of F around closed curves?

• A. ∮_C F·dr = 1 for all closed curves (unit circulation)
• B. ∮_C F·dr = 0 for any closed curve in the domain
• C. The field has no flux through any surface
• D. The curl equals the divergence everywhere

Answer: (B)

If ∇×F = 0, Stokes gives ∮_C F·dr = ∬_S 0·n dS = 0. An irrotational field has zero circulation around every closed curve, which on simply connected domains is equivalent to the field being conservative and path-independent.

Question 3

Stokes' theorem gives the same value for ∬_S (∇×F)·n dS regardless of which spanning surface S you choose, as long as S is bounded by the same closed curve C.

• T. True
• F. False

Answer: True

Surface-independence is the theorem's deep content. It follows from ∇·(∇×F) = 0 always. Any two surfaces spanning C together form a closed surface; by the divergence theorem, the net flux of ∇×F through a closed surface equals the volume integral of ∇·(∇×F) — which is zero. So the flux through any two spanning surfaces is equal.

Question 4

Stokes' theorem is a mostly separate result from Green's theorem, with no mathematical relationship between them.

• T. True
• F. False

Answer: False

Green's theorem is a special case of Stokes' theorem applied to a flat region in ℝ². When the surface S is a planar region D and the boundary curve C lies in the plane, Stokes reduces exactly to Green's theorem. Both are instances of the same master result: ∫_∂Ω ω = ∫_Ω dω.

Question 5

Explain why Stokes' theorem gives the same answer regardless of which spanning surface S you choose for a given boundary curve C.

Model answer: Because the curl is always divergence-free (∇·(∇×F) = 0). Any two different spanning surfaces S₁ and S₂ with the same boundary C together form a closed surface. By the divergence theorem, the net flux of ∇×F through a closed surface equals the volume integral of ∇·(∇×F) over the enclosed region — which is zero. So the flux through S₁ equals the flux through S₂.

The algebraic identity div(curl) = 0 is what makes the theorem strategically useful: you can freely choose whichever spanning surface simplifies the computation, guaranteed that the answer is independent of that choice.