Subtopic: Vector differential operators – del, gradient
Question Statement:
The maximum rate of increase of the scalar field $\phi(x, y, z) = 2x^2 + 3y^2 + z^2$ at the point $(1, 1, 2)$ is ________ (round off to 2 decimal places).
Answer Key:
8.25
Step-by-Step Solution & Explanation
The maximum rate of increase of a scalar field $\phi$ at any point occurs in the direction of the gradient $\nabla \phi$, and its magnitude equals $|\nabla \phi|$.
Compute the gradient:
$\nabla \phi = \frac{\partial \phi}{\partial x}\hat{i} + \frac{\partial \phi}{\partial y}\hat{j} + \frac{\partial \phi}{\partial z}\hat{k} = (4x)\hat{i} + (6y)\hat{j} + (2z)\hat{k}$
At the point $(1, 1, 2)$:
$\nabla \phi = 4(1)\hat{i} + 6(1)\hat{j} + 2(2)\hat{k} = 4\hat{i} + 6\hat{j} + 4\hat{k}$
Magnitude:
$|\nabla \phi| = \sqrt{4^2 + 6^2 + 4^2} = \sqrt{16 + 36 + 16} = \sqrt{68} \approx 8.25$