Question

The directional derivative of a function f (x, y, z) at some point is greatest in the direction parallel to i + j − k. In this direction, the value of the directional derivative is 2√3.

(a) What is ∇f at this point? Give reasons for your answer.

(b) What is the directional derivative of f at the point in the direction 1√2 (i + j)?

Answer 1

(a) This problem asks us to reconstruct a vector given its direction and magnitude. We know that the directional derivative is greatest in the direction of the gradient. The unit vector representing this direction is

d = 1√3 (i + j − k)

Thus d is the direction of the gradient. Regarding its magnitude, we have the following rule: The directional derivative in the direction of the gradient equals its norm. In other words, if d points along the gradient, then

Why is this the case? If d points in the direction of the gradient, then

and therefore

We know the magnitude of the directional derivative and thus the norm of the gradient,

(b) The unit vector representing the given direction is

d = 1 √2 (i + j) ,

and the directional derivative is

∇f · d = 2√2 .