Questions: Curl and Divergence of Vector Fields

Question 1

At a point in a 3D vector field representing fluid flow, the divergence is large and positive. What does this mean physically?

• A. The fluid is spinning rapidly counterclockwise at that point
• B. The fluid velocity is large in magnitude at that point
• C. The point acts like a source — nearby flow vectors point outward, as if fluid were being emitted there
• D. The field is conservative at that point, meaning the flow is path-independent

Answer: (C)

Divergence measures net outflow: positive divergence at a point means nearby vectors point away from that region, like a source emitting fluid. It says nothing about rotation (that's curl) or about the magnitude of the velocity. A large divergence means a strong source, not fast flow. Imagine enclosing the point in a tiny balloon — positive divergence means the balloon inflates. Zero divergence everywhere means the fluid is incompressible: as much flows in as out at every point.

Question 2

A vector field F is known to be conservative. What must be true about curl(F)?

• A. curl(F) must be a large positive constant, since conservative fields store energy
• B. curl(F) = 0 everywhere — conservative fields are irrotational
• C. curl(F) equals the gradient of the potential function associated with F
• D. curl(F) may be nonzero, but its integral over any closed surface must be zero

Answer: (B)

For a conservative field F on a simply connected domain, curl(F) = 0 everywhere — the field is irrotational. This is equivalent to F being path-independent (the line integral between any two points doesn't depend on the path taken). Intuitively, a conservative field has no local rotation — a paddle wheel placed anywhere in the flow would not spin. The converse is also true on simply connected domains: curl(F) = 0 implies F is conservative.

Question 3

A vector field with zero divergence everywhere should be conservative (path-independent).

• T. True
• F. False

Answer: False

Zero divergence (∇ · F = 0) means the field is solenoidal or incompressible — as much flows in as out at every point. This says nothing about rotation or path-independence. Conservatism requires zero curl (∇ × F = 0), not zero divergence. These are completely different properties. For example, the magnetic field has zero divergence everywhere (no magnetic monopoles), but is not generally conservative — it can have nonzero curl (related to electric currents by Ampere's law).

Question 4

The identity curl(∇f) = 0 means that the curl of any gradient field is always zero — gradient fields are always irrotational.

• T. True
• F. False

Answer: True

This is a fundamental vector calculus identity: for any scalar function f with continuous second partial derivatives, ∇ × (∇f) = 0. Geometrically, this makes sense: a gradient field points in the direction of steepest increase of f, and such a field has no local rotation — a paddle wheel placed in it would not spin. This identity is why potential energy functions work: the force F = −∇U has curl(F) = −curl(∇U) = 0, confirming it is conservative.

Question 5

Using the analogy of fluid flow, explain what divergence and curl each measure. What physical question does each operator answer at a point in the field?

Model answer: Divergence answers: is this point a source or a sink? Positive divergence means fluid flows outward from the region (like a drain running in reverse); negative means fluid converges inward (like a drain). Zero divergence means the fluid is incompressible — volume is conserved. Curl answers: is there local rotation? The curl vector points along the axis of rotation and its magnitude gives the spin rate — a paddle wheel placed at the point would spin if curl is nonzero, and not spin if curl is zero.

These two operators extract completely different geometric information from the same vector field. A field can have nonzero divergence and zero curl (pure source/sink, no rotation), zero divergence and nonzero curl (incompressible but rotating, like an ideal vortex), both nonzero, or both zero. Understanding which property each operator measures prevents the common confusion of mixing them up when reasoning about physical fields like fluid flow, electric fields, or magnetic fields.