Questions: Curl and Divergence of Vector Fields

Question 1

For a 2D vector field F = ⟨P, Q⟩, which expression gives the z-component of the curl (the scalar 2D curl)?

• A. ∂P/∂x + ∂Q/∂y
• B. ∂Q/∂x − ∂P/∂y
• C. ∂P/∂y − ∂Q/∂x
• D. ∂P/∂x − ∂Q/∂y

Answer: (B)

The 2D curl is ∂Q/∂x − ∂P/∂y — the cross-partial pattern with a minus sign. This is a common confusion point: the divergence uses the same partial derivatives but with a plus sign (∂P/∂x + ∂Q/∂y). Curl involves the 'cross' combination of partials, while divergence involves the 'straight' combination.

Question 2

A vector field with zero divergence everywhere has no sources or sinks — fluid neither accumulates nor drains at any point.

• T. True
• F. False

Answer: True

Divergence measures net outflow per unit volume at a point. Zero divergence means the field is divergence-free (or incompressible): as much flows into any region as flows out. This is exactly the condition of having no sources (positive divergence) or sinks (negative divergence). Incompressible fluid flow satisfies ∇ · F = 0 everywhere.

Question 3

What does it mean physically if the curl of a vector field F is zero at every point in its domain?

Model answer: The field is irrotational — there is no local rotation or swirling tendency at any point. On a simply connected domain, this is equivalent to F being a conservative field (having a scalar potential function).

Curl measures the rotation/circulation tendency of a field. Imagine placing a tiny paddle wheel at a point: if curl F = 0, the wheel won't spin. Zero curl (irrotational) plus a simply connected domain guarantees F = ∇f for some scalar f — a conservative field where line integrals are path-independent.