Question Bank • QB_EM_VC_007 NAT • 2 Marks

Section 1: Engineering Mathematics

Vector Calculus

Subtopic: Divergence and curl

Question Statement:

The divergence of the vector field $\vec{F} = (x^2 y)\hat{i} + (y^2 z)\hat{j} + (z^2 x)\hat{k}$ at the point $(1, 2, 3)$ is ________ (answer in integer).

Answer Key:

22

Step-by-Step Solution & Explanation

The divergence of $\vec{F} = F_1\hat{i} + F_2\hat{j} + F_3\hat{k}$ is: $\nabla \cdot \vec{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z}$ $\frac{\partial}{\partial x}(x^2 y) = 2xy$ $\frac{\partial}{\partial y}(y^2 z) = 2yz$ $\frac{\partial}{\partial z}(z^2 x) = 2zx$ At $(1, 2, 3)$: $\nabla \cdot \vec{F} = 2(1)(2) + 2(2)(3) + 2(3)(1) = 4 + 12 + 6 = 22$
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