Vector Calculus: Match Differential/Integral Conditions to Field Properties List I A. ∇ × F = 0 B. ∇ · F = 0 C. ∮ F · dl = 0 List II Irrotational vector Conservative field Solenoidal vector

Introduction / Context:
This matching item checks core vector-calculus relationships used in electromagnetics and fluid mechanics. Recognizing when a vector field is irrotational, solenoidal, or conservative is essential for applying Maxwell’s equations, potential theory, and circulation/flux theorems.

Given Data / Assumptions:

• A: Curl of the field equals zero.
• B: Divergence of the field equals zero.
• C: Line integral of the field around any closed path equals zero.
• Assume simply connected regions when linking irrotational and conservative properties.

Concept / Approach:

Definitions connect operators to properties: curl F = 0 implies an irrotational field; div F = 0 implies a solenoidal (divergence-free) field; a vanishing closed-loop line integral for every loop implies a conservative field (path independence), which in a simply connected domain is equivalent to curl F = 0 and existence of a scalar potential.

Step-by-Step Solution:

Verification / Alternative check:

By Stokes’ theorem, ∮ F · dl = ∬ (∇ × F) · n dS. If this integral is zero for every surface bounded by the loop, then ∇ × F must vanish everywhere (in a simply connected region), confirming conservative behavior. Gauss’s theorem connects divergence to net flux through closed surfaces; zero divergence gives zero net outflow, i.e., solenoidal.

Why Other Options Are Wrong:

Mapping B to “irrotational” confuses curl with divergence. Mapping A to “conservative” skips the simply connected requirement explicitly tied to the integral condition. Swapping 2 and 3 reverses the meanings of conservative and solenoidal.

Common Pitfalls:

Forgetting the topological caveat: ∇ × F = 0 does not imply conservative in multiply connected regions unless the loop integral condition also holds. Similarly, solenoidal does not imply irrotational.

Final Answer:

A-1, B-2, C-3