Question Bank • QB_EM_VC_003 MCQ • 1 Mark

Section 1: Engineering Mathematics

Vector Calculus

Subtopic: Scalar and vector point functions

Question Statement:

A vector point function $\vec{F}$ is called conservative (irrotational) in a simply connected domain if and only if:

Options:

  • (A) $\nabla \times \vec{F} = \vec{0}$
  • (B) $\nabla \cdot \vec{F} = 0$
  • (C) $\nabla^2 \vec{F} = \vec{0}$
  • (D) $\vec{F} \cdot d\vec{r} = 0$ along any open curve

Answer Key:

A

Step-by-Step Solution & Explanation

A vector field $\vec{F}$ is conservative if its curl vanishes everywhere ($\nabla \times \vec{F} = \vec{0}$). This is the necessary and sufficient condition for the existence of a single-valued scalar potential function $\phi$ such that $\vec{F} = \nabla \phi$, meaning line integrals are path-independent.
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