Allied & ICAR Exam Bank • ALLIED_2670 NAT • 2 Marks

Section 1: Engineering Mathematics

Vector Calculus

Subtopic: Vector differentiation

Question Statement:

For the scalar field $\phi(x, y, z) = x^2 y + y^2 z + z^2 x$, the magnitude of $\nabla \phi$ at the point $(1, 1, 1)$ is $k \sqrt{3}$. The value of $k$ is:

Answer Key:

3

Step-by-Step Solution & Explanation

$\nabla \phi = (2xy + z^2)\hat{i} + (x^2 + 2yz)\hat{j} + (y^2 + 2zx)\hat{k}$. At $(1, 1, 1)$, $\nabla \phi = 3\hat{i} + 3\hat{j} + 3\hat{k}$. Magnitude = $\sqrt{3^2 + 3^2 + 3^2} = \sqrt{27} = 3\sqrt{3}$. Hence $k = 3$.
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