Directional Derivatives — Question 10

Question 10 – Solution

Strategy. Resolve the gradient into components parallel and perpendicular to $a$ and compare symmetric directions.

Step 1: Decompose Let $b$ be a unit vector perpendicular to $a$, and write $$\nabla f=Aa+Bb.$$ The two unit directions at angles $\pm\pi/3$ from $a$ are $$u_\pm=\frac 12a\pm\frac{\sqrt 3}{2}b.$$

Step 2: Use both measurements $$3=\nabla f\cdot u_\pm=\frac A2\pm\frac{\sqrt 3 B}{2}.$$ Adding gives $6=A$, while subtracting gives $B=0$.

Step 3: Result Therefore $$\boxed{\nabla f(P)=6a}.$$ The greatest directional derivative is the gradient magnitude, $$\boxed{D_{\max}f(P)=6},$$ attained uniquely in direction $a$. Checking either prescribed direction gives $6\cos(\pi/3)=3$.