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QB_EM_ADV_048
Question Bank • QB_EM_ADV_048
MSQ • 2 Marks
Section 1: Engineering Mathematics
Vector Calculus
Subtopic: Divergence and curl
Question Statement:
Which of the following vector identities is/are mathematically VALID for any sufficiently smooth scalar field $\phi$ and vector fields $\mathbf{A}, \mathbf{B}$?
Options:
(A) $\nabla \times (\nabla \phi) = \mathbf{0}$
(B) $\nabla \cdot (\nabla \times \mathbf{A}) = 0$
(C) $\nabla \times (\nabla \times \mathbf{A}) = \nabla(\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}$
(D) $\nabla \cdot (\mathbf{A} \times \mathbf{B}) = \mathbf{B} \cdot (\nabla \times \mathbf{A}) - \mathbf{A} \cdot (\nabla \times \mathbf{B})$
Step-by-Step Solution & Explanation
1. **Curl of Gradient (Option A)**: For any $C^2$ scalar field $\phi$, $\nabla \times (\nabla \phi) = \mathbf{0}$ due to the equality of mixed partial derivatives (True).
2. **Divergence of Curl (Option B)**: The divergence of any curl field is identically zero: $\nabla \cdot (\nabla \times \mathbf{A}) = 0$ (True).
3. **Vector Laplacian Identity (Option C)**: $\nabla \times (\nabla \times \mathbf{A}) = \nabla(\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}$ is a cornerstone identity in wave and electromagnetics theory (True).
4. **Divergence of Cross Product (Option D)**: Product expansion gives $\nabla \cdot (\mathbf{A} \times \mathbf{B}) = \mathbf{B} \cdot (\nabla \times \mathbf{A}) - \mathbf{A} \cdot (\nabla \times \mathbf{B})$ (True).
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