A fluid velocity field F has ∇·F > 0 at a point P. What does this tell you about the flow at P?
AThe fluid is spinning at P — divergence measures rotation
BThere is a source at P — fluid is flowing outward, expanding away from this point
CThe fluid flow is conservative at P
DThe field has no curl at P
Divergence ∇·F measures the net rate of outward flow (expansion) at a point. ∇·F > 0 means fluid is being created there — a source. ∇·F < 0 would be a sink. Rotation is measured by curl, not divergence — this is the most common confusion between the two operators.
Question 2 Multiple Choice
A vector field F has ∇×F = 0 everywhere. Which of the following must be true?
AF is the zero vector field
BF has no sources or sinks (∇·F = 0 everywhere)
CF can be written as the gradient of some scalar potential φ
DF is a constant vector field
∇×F = 0 means F is irrotational, which (on simply connected domains) is equivalent to F being a conservative field — expressible as F = ∇φ for some scalar potential. This says nothing about divergence: a conservative field can have sources and sinks. The zero field satisfies both conditions, but many non-zero, non-constant fields have zero curl.
Question 3 True / False
The divergence of the curl of any smooth vector field is always zero: ∇·(∇×F) = 0.
TTrue
FFalse
Answer: True
This identity follows from the antisymmetry of the cross product and the symmetry of mixed partial derivatives. Intuitively: if curl measures local rotation, divergence of a curl would measure 'net outflow of rotation,' which is geometrically zero — rotation has no net source or sink. This identity is essential background for Stokes' theorem and the Divergence Theorem.
Question 4 True / False
If a vector field F has ∇·F = 0 everywhere (incompressible), then F is conservative.
TTrue
FFalse
Answer: False
Incompressible (zero divergence) and conservative (zero curl) are entirely different conditions. ∇·F = 0 means no sources or sinks — as much flows in as flows out. ∇×F = 0 means no local rotation, which implies the field is a gradient field. Neither condition implies the other. For example, a steady vortex flow can be incompressible but highly rotational (nonzero curl).
Question 5 Short Answer
Explain why curl is a vector while divergence is a scalar, and what each one measures about a vector field at a point.
Think about your answer, then reveal below.
Model answer: Divergence sums the self-derivatives of each component (∂P/∂x + ∂Q/∂y + ∂R/∂z), producing a single number that measures net outward flow — one scalar per point. Curl computes cross-derivatives between different components, producing a vector whose direction indicates the axis of rotation and whose magnitude measures the angular speed of local spinning.
The scalar vs. vector distinction reflects the geometry. Outflow has no preferred direction — it's symmetric — so divergence is scalar. Rotation has an axis and sense (clockwise vs. counterclockwise), so curl must be a vector to encode that directional information. This also explains why the del notation works: ∇·F is a dot product (scalar), ∇×F is a cross product (vector).