Allied & ICAR Exam Bank • ALLIED_2689
MSQ • 2 Marks
Section 1: Engineering Mathematics
Vector Calculus
Subtopic: Physical interpretations – line, surface and volume integrals
Question Statement:
For a conservative force field $\vec{F} = \nabla \phi$, which of the following statements are TRUE?
Options:
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(A) The line integral $\int_A^B \vec{F} \cdot d\vec{r}$ depends only on the endpoints $A$ and $B$
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(B) $\nabla \times \vec{F} = 0$
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(C) $\oint_C \vec{F} \cdot d\vec{r} = 0$ for all simple closed contours $C$
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(D) $\nabla \cdot \vec{F} = 0$ always
Step-by-Step Solution & Explanation
(A), (B), (C) are equivalent conditions for conservative fields. (D) is false because $\nabla \cdot \vec{F} = \nabla^2 \phi$, which need not be zero.
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