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QB_EM_ADV_045
Question Bank • QB_EM_ADV_045
NAT • 2 Marks
Section 1: Engineering Mathematics
Vector Calculus
Subtopic: Vector differential operators – del, gradient
Question Statement:
The directional derivative of $\phi(x, y, z) = x^2 y z + 4 x z^2$ at the point $P(1, -2, -1)$ in the direction of the vector $\mathbf{a} = 2\hat{i} - \hat{j} - 2\hat{k}$ is ________ (round off to two decimal places).
Step-by-Step Solution & Explanation
Compute the gradient of $\phi$:
$\nabla \phi = \left( 2xyz + 4z^2 \right)\hat{i} + \left( x^2 z \right)\hat{j} + \left( x^2 y + 8xz \right)\hat{k}$
Evaluate at $P(1, -2, -1)$:
$\frac{\partial \phi}{\partial x} = 2(1)(-2)(-1) + 4(-1)^2 = 4 + 4 = 8$
$\frac{\partial \phi}{\partial y} = (1)^2 (-1) = -1$
$\frac{\partial \phi}{\partial z} = (1)^2 (-2) + 8(1)(-1) = -2 - 8 = -10$
$\nabla \phi(1, -2, -1) = 8\hat{i} - \hat{j} - 10\hat{k}$
The unit vector along $\mathbf{a} = 2\hat{i} - \hat{j} - 2\hat{k}$ is:
$\hat{a} = \frac{2\hat{i} - \hat{j} - 2\hat{k}}{\sqrt{2^2 + (-1)^2 + (-2)^2}} = \frac{2\hat{i} - \hat{j} - 2\hat{k}}{3}$
Directional derivative:
$D_{\hat{a}} \phi = \nabla \phi \cdot \hat{a} = \frac{8(2) + (-1)(-1) + (-10)(-2)}{3} = \frac{16 + 1 + 20}{3} = \frac{37}{3} \approx 12.33$
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