Question Bank • QB_EM_146 NAT • 2 Marks

Section 1: Engineering Mathematics

Vector Calculus

Subtopic: Vector differential operators – del, gradient

Question Statement:

Calculate the directional derivative of the scalar field $\phi(x, y, z) = x^2 y + y z$ at the point $P(1, 2, 1)$ in the direction of the unit vector $\hat{u} = \frac{1}{\sqrt{3}}(\hat{i} + \hat{j} + \hat{k})$ (round off to 2 decimal places).

Answer Key:

4.62

Step-by-Step Solution & Explanation

Gradient of $\phi$: $\nabla \phi = \left(2xy\right)\hat{i} + \left(x^2 + z\right)\hat{j} + y\hat{k}$ At point $P(1, 2, 1)$: $\nabla \phi = (2 \times 1 \times 2)\hat{i} + (1^2 + 1)\hat{j} + 2\hat{k} = 4\hat{i} + 2\hat{j} + 2\hat{k}$ Directional derivative: $D_{\hat{u}}\phi = \nabla \phi \cdot \hat{u} = \frac{4(1) + 2(1) + 2(1)}{\sqrt{3}} = \frac{8}{\sqrt{3}} \approx \frac{8}{1.732} \approx 4.619 \approx 4.62$ Acceptable range: $4.5 - 4.7$.
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