The rate of increase, of a scalar field f(x,y,z)=xyz, in the direction v=(2,1,2) at a point (0,2,1) is

GATE 2023 EC – Question 12

Engineering Mathematics · Calculus · 1 mark · Multiple choice

The rate of increase, of a scalar field $f(x,y,z)=xyz$, in the direction $\boldsymbol v=(2,1,2)$ at a point $(0,2,1)$ is

$\dfrac23$
$\dfrac43$
2
4

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Correct answer: (B) $\dfrac43$

Explanation

$\nabla f=(yz,\,xz,\,xy)$ which at $(0,2,1)$ is $(2,0,0)$. The unit vector along $\boldsymbol v$ is $\frac13(2,1,2)$. The directional derivative is $(2,0,0)\cdot\frac13(2,1,2)=\frac43$.