Subtopic: Vector differential operators – del, gradient
Question Statement:
A unit vector normal to the level surface $\phi(x, y, z) = c$ at a given point $P$ is given by:
Options:
(A) $\frac{\nabla \phi}{|\nabla \phi|}$
(B) $\nabla \times (\nabla \phi)$
(C) $\nabla(\nabla \cdot \phi)$
(D) $\frac{\nabla \cdot \phi}{|\nabla \phi|}$
Answer Key:
A
Step-by-Step Solution & Explanation
The gradient vector $\nabla \phi$ is always perpendicular (normal) to the level surface $\phi(x, y, z) = c$ at every point. Dividing by its Euclidean norm $|\nabla \phi|$ yields the unit normal vector $\hat{n} = \frac{\nabla \phi}{|\nabla \phi|}$.