Questions: Stokes' Theorem and the Divergence Theorem

Question 1

Computing ∬_S F · dS directly over a complicated closed surface S is very difficult. However, the enclosed volume V has a simple shape, and ∇ · F = 4 everywhere inside V (volume of V = 6). Which theorem applies, and what is the flux?

• A. Stokes' theorem; compute the curl of F and integrate over a bounding curve instead.
• B. The Divergence theorem; flux = ∭_V ∇ · F dV = 4 × 6 = 24.
• C. The Divergence theorem; flux = ∬_S 4 dS, still requiring the surface integral.
• D. The Fundamental Theorem of Calculus, since F has constant divergence.

Answer: (B)

The Divergence theorem converts the surface integral of F · dS over a closed surface into the volume integral of ∇ · F over the enclosed region: ∬_S F · dS = ∭_V ∇ · F dV. With ∇ · F = 4 everywhere and volume = 6, the answer is 24. Option C is the error of knowing the theorem's name but not applying it — the whole point is to *replace* the surface integral with the volume integral.

Question 2

According to Stokes' theorem, two different oriented surfaces S₁ and S₂ both have the same closed curve C as their boundary. Which statement is correct?

• A. ∬_{S₁} (∇ × F) · dS and ∬_{S₂} (∇ × F) · dS may differ if the surfaces have different areas.
• B. ∬_{S₁} (∇ × F) · dS = ∬_{S₂} (∇ × F) · dS, because both equal the same line integral ∮_C F · dr.
• C. ∬_{S₁} (∇ × F) · dS = ∬_{S₂} (∇ × F) · dS only if F is conservative.
• D. You must use the flat surface to apply Stokes' theorem; curved surfaces give different results.

Answer: (B)

Stokes' theorem equates the line integral ∮_C F · dr with the surface integral of the curl over any surface bounded by C. Since both S₁ and S₂ share the same boundary curve C, both surface integrals equal the same line integral — they must be equal. This surface-independence is one of Stokes' most powerful features: you can choose whichever surface makes the computation easiest.

Question 3

The Divergence theorem, Stokes' theorem, and the Fundamental Theorem of Calculus are all instances of the same pattern: an integral over a region's interior equals an integral over its boundary.

• T. True
• F. False

Answer: True

Yes. The Fundamental Theorem says ∫_a^b f′ dx = f(b) − f(a): integral of derivative over an interval equals function evaluated on the boundary (two points). Green's theorem, Stokes' theorem, and the Divergence theorem are the 2D and 3D versions of this pattern. In each case, an integral involving a differential operator (curl, divergence) over the interior equals an integral over the boundary (curve, surface).

Question 4

The Divergence theorem converts a surface integral into a line integral by integrating the divergence of F along the boundary curve of the surface.

• T. True
• F. False

Answer: False

The Divergence theorem converts a *surface* integral (flux through a closed surface) into a *volume* integral (of divergence over the enclosed 3D region). It has nothing to do with line integrals. That confusion conflates it with Stokes' theorem, which connects a line integral to a surface integral of the curl. These are different theorems: Stokes relates line ↔ surface; Divergence relates surface ↔ volume.

Question 5

If ∇ · F = 0 everywhere in a region, what does the Divergence theorem tell you about the flux through any closed surface in that region? Why does this make physical sense?

Model answer: The Divergence theorem gives ∬_S F · dS = ∭_V ∇ · F dV = 0. Physically, zero divergence means there are no sources or sinks — the field neither originates nor terminates at any interior point. So every field line that enters a closed surface must also exit it: the net outward flux is zero. This is exactly Gauss's law for electric fields in a charge-free region, or the incompressibility condition for a fluid with no sources.

The physical interpretation of divergence as 'source density' is key. When ∇ · F = 0, the vector field is source-free, and by conservation, the total flow through any closed surface must balance — what goes in must come out.