Two different surfaces S₁ and S₂ share the same oriented boundary curve C. The vector field F has ∇×F ≠ 0 throughout space. What can you conclude?
A∬_{S₁}(∇×F)·dS = ∬_{S₂}(∇×F)·dS, because both equal the same boundary circulation ∮_C F·dr
BThe two integrals are different, because the curl field varies in space and different surfaces sample different regions
CThe two integrals are equal only if S₁ and S₂ have the same area
DThe two integrals are equal only if F is conservative
Stokes' theorem says ∬_S (∇×F)·dS = ∮_C F·dr for any surface S spanning C. Since both S₁ and S₂ share boundary C, both curl-flux integrals equal the same line integral ∮_C F·dr — even though ∇×F ≠ 0 and the surfaces sample different regions of space. This is the deep result: the flux of curl through a surface depends only on the boundary, not on which spanning surface is chosen.
Question 2 Multiple Choice
You know ∇×F = 0 everywhere in a simply-connected region, and C is a closed loop bounding a surface S in that region. Which reasoning correctly concludes that ∮_C F·dr = 0?
AThe argument is circular — you need to verify F has a potential function before applying this reasoning
B∬_S (∇×F)·dS = ∬_S 0 dS = 0, and by Stokes' theorem this equals ∮_C F·dr, so the circulation is zero
CStokes' theorem doesn't apply here because ∇×F = 0 means there is no curl field to integrate
DThe circulation is zero only if C is a circle — other loop shapes require a different argument
This is a direct application of Stokes' theorem. Because ∇×F = 0 everywhere, the surface integral ∬_S (∇×F)·dS = 0 regardless of which surface S you choose. Stokes' theorem then forces ∮_C F·dr = 0. Option A confuses the logical direction: Stokes' theorem is the tool that proves the circulation vanishes; you don't need to find a potential function first. The curl condition ∇×F = 0 is doing all the work.
Question 3 True / False
Stokes' theorem implies that the flux of the curl through a surface depends only on the boundary curve of that surface, not on which particular surface spanning that curve you choose.
TTrue
FFalse
Answer: True
This is the core geometric content of Stokes' theorem. For any two surfaces sharing the same oriented boundary C, both curl-flux integrals equal ∮_C F·dr — the same quantity. So the value is entirely determined by C. This has a powerful consequence: to evaluate ∬_S (∇×F)·dS, you can replace S with any other convenient surface that has the same boundary, often dramatically simplifying the computation.
Question 4 True / False
Stokes' theorem states that the circulation of F around a curve C equals the flux of ∇×F through C itself.
TTrue
FFalse
Answer: False
C is a curve — you cannot integrate a vector field over a curve using a surface integral. Stokes' theorem states that circulation around C equals the flux of ∇×F through a surface S whose *boundary* is C: ∮_C F·dr = ∬_S (∇×F)·dS. The surface S is a 2D object bounded by C; it is distinct from C itself. Confusing the boundary curve with the spanning surface is a common sign of misunderstanding the theorem's structure.
Question 5 Short Answer
Explain in your own words how the tiling argument derives Stokes' theorem. Why does only the outer boundary survive when you sum the circulation contributions from all the tiny patches?
Think about your answer, then reveal below.
Model answer: Divide the surface S into many tiny parallelogram patches. Each patch has a small boundary loop, and the circulation around that tiny loop is approximately (∇×F)·n̂ ΔA — the local curl dotted with the patch's normal times its area. Now sum all these tiny circulations. Every interior edge is shared by exactly two adjacent patches, and those patches traverse the shared edge in opposite directions (one clockwise, one counterclockwise relative to that edge). These contributions cancel exactly. The only edges that are not canceled are those on the outer boundary of the entire surface — each of those is traversed only once. The surviving sum is therefore the line integral ∮_C F·dr around the outer boundary. Making this argument rigorous gives Stokes' theorem: the global circulation equals the sum of local curl contributions.