The directional derivative of a scalar field $\phi(x, y, z)$ at a point $P$ attains its maximum value along:
Options:
(A) The tangent to the level surface $\phi = c$
(B) The direction of the gradient vector $\nabla \phi$
(C) Any direction perpendicular to $\nabla \phi$
(D) The direction opposite to $\nabla \phi$
Answer Key:
B
Step-by-Step Solution & Explanation
The directional derivative along any unit vector $\hat{u}$ is given by:
$D_{\hat{u}} \phi = \nabla \phi \cdot \hat{u} = |\nabla \phi| \cos\theta$
Since $-1 \le \cos\theta \le 1$, this quantity is strictly maximized when $\cos\theta = 1$ (i.e., $\theta = 0$), meaning $\hat{u}$ points in the exact direction of $\nabla \phi$, with maximum value equal to $|\nabla \phi|$.