Allied & ICAR Exam Bank • ALLIED_0045 NAT • 2 Marks

Section 1: Engineering Mathematics

Vector Calculus

Subtopic: Vector differential operators – del, gradient

Question Statement:

The directional derivative of the scalar field $\phi(x, y, z) = x^2 y + y z^2$ at $P(1, 2, -1)$ in the direction of $\vec{v} = 2\hat{i} + \hat{j} - 2\hat{k}$ is:

Answer Key:

6

Step-by-Step Solution & Explanation

$\nabla \phi = 2xy\hat{i} + (x^2 + z^2)\hat{j} + 2yz\hat{k} = 4\hat{i} + 2\hat{j} - 4\hat{k}$. Unit vector $\hat{u} = (2\hat{i} + \hat{j} - 2\hat{k})/3$. Directional derivative $= (4(2) + 2(1) - 4(-2))/3 = (8 + 2 + 8)/3 = 6.0$.
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