Question Bank • QB_EM_092 NAT • 2 Marks

Section 1: Engineering Mathematics

Vector Calculus

Subtopic: Scalar and vector point functions

Question Statement:

Calculate the directional derivative of $\phi(x, y, z) = 2xy + z^2$ at the point $P(1, 2, 3)$ in the direction of the vector $\vec{v} = 3\hat{i} + 4\hat{j}$.

Answer Key:

4

Step-by-Step Solution & Explanation

Gradient $\nabla \phi = \frac{\partial \phi}{\partial x}\hat{i} + \frac{\partial \phi}{\partial y}\hat{j} + \frac{\partial \phi}{\partial z}\hat{k}$: $\nabla \phi = (2y)\hat{i} + (2x)\hat{j} + (2z)\hat{k}$ At $P(1, 2, 3)$: $\nabla \phi = 2(2)\hat{i} + 2(1)\hat{j} + 2(3)\hat{k} = 4\hat{i} + 2\hat{j} + 6\hat{k}$ Unit vector in direction of $\vec{v}$: $\hat{u} = \frac{3\hat{i} + 4\hat{j}}{\sqrt{3^2 + 4^2}} = \frac{3}{5}\hat{i} + \frac{4}{5}\hat{j}$ Directional derivative: $D_{\hat{u}} = \nabla \phi \cdot \hat{u} = \left(4 \times \frac{3}{5}\right) + \left(2 \times \frac{4}{5}\right) + (6 \times 0) = \frac{12 + 8}{5} = \frac{20}{5} = 4.0$
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