You need to compute the outward flux of F = ⟨x², y², z²⟩ through the unit sphere. A classmate proposes carefully parametrizing the sphere and computing the surface integral directly. What does the divergence theorem offer instead?
AConvert the surface integral to a line integral along the sphere's equatorial circle.
BCompute ∇·F = 2x + 2y + 2z and integrate this over the unit ball using a triple integral.
CReplace F with a simpler field that has the same divergence at the center of the sphere.
DNothing — the divergence theorem only applies to fields with zero curl.
The divergence theorem trades the surface integral for ∭_W ∇·F dV. Here ∇·F = 2x + 2y + 2z, and by symmetry the integrals of 2x, 2y, and 2z over the unit ball all equal zero, so the flux is 0. This is far simpler than parametrizing the sphere. Option D is a common misconception — the divergence theorem has no curl restriction; it applies to any smooth vector field over a region bounded by a closed surface.
Question 2 Multiple Choice
The divergence theorem states ∬_S F·n dS = ∭_W ∇·F dV. If ∇·F > 0 throughout a region W, what does this imply about the net flux through its boundary S?
ANet flux through S is zero — sources and sinks cancel inside W.
BNet flux through S is negative — positive divergence pushes the field inward.
CNet flux through S is positive — sources inside W produce outward flow through S.
DThe theorem says nothing about flux direction; it only gives the magnitude.
Positive divergence means sources inside W are expanding the field outward. By the theorem, the flux integral equals the volume integral of divergence — which is positive if divergence is positive throughout. So more fluid (or field) exits S than enters, giving positive net outward flux. Option B confuses the direction: positive divergence in the interior creates positive outward flux, not inward.
Question 3 True / False
For the vector field F = ⟨x, y, z⟩, the outward flux through any closed surface bounding a region W equals three times the volume of W.
TTrue
FFalse
Answer: True
∇·F = ∂x/∂x + ∂y/∂y + ∂z/∂z = 1 + 1 + 1 = 3. By the divergence theorem, ∬_S F·n dS = ∭_W 3 dV = 3·Vol(W). No surface parametrization is needed at all — the constant divergence makes the volume integral trivial. This is one of the most elegant demonstrations of the theorem's computational power.
Question 4 True / False
The divergence theorem can mainly be used in one direction: to convert surface integrals into volume integrals, not the reverse.
TTrue
FFalse
Answer: False
The theorem is an equality, so it works in both directions. Sometimes a volume integral over a complicated region W is easier to evaluate after converting it to a surface integral over the simpler boundary ∂W. The practical strategy is to choose whichever side of the equation is easier to compute. This bidirectional flexibility — volume ↔ surface — is a defining feature of all the fundamental theorems of vector calculus.
Question 5 Short Answer
Using the analogy of fluid flow, explain the physical intuition behind the divergence theorem.
Think about your answer, then reveal below.
Model answer: If a region W is filled with fluid whose velocity is F, then ∬_S F·n dS counts the net volume of fluid per unit time flowing out through the boundary S. ∇·F at each interior point measures how strongly the fluid is expanding there (a source is positive, a sink is negative). The divergence theorem says these two perspectives must agree: the total outflow through the boundary equals the total source strength accumulated throughout the interior. It is an exact bookkeeping identity.
The intuition is conservation: whatever net fluid exits the boundary must have been generated inside. If there are no sources or sinks (∇·F = 0 everywhere — a divergence-free field), the net flux through any closed surface is zero. This physical picture connects to the Fundamental Theorem of Calculus: boundary information and interior information are two sides of the same accounting ledger.