ENGINEERING MATHEMATICS · VECTOR CALCULUS

GATE 2023 EC, question 12: Vector calculus — solution and answer

GATE 2023 · ECQ.12 · 1 mark · MCQ Checked against the official key

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

(A)
$\frac{2}{3}$
(B)
$\frac{4}{3}$
(C)
$2$
(D)
$4$

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Answer and worked solution

ANSWER

The rate of increase of $f = x y z$ at $(0, 2, 1)$ in the direction $(2, 1, 2)$ is $\frac{4}{3}$ , option (B).

Official key: B · Official answer key, GATE 2023 EC (IIT Kanpur)

The rate of increase of $f$ in a given direction is the directional derivative $D_{\overset{v}{^}} f = \nabla f \cdot \overset{v}{^}$ , where $\overset{v}{^}$ is the unit vector along $v$ .
Gradient of $f = x y z$ : $\nabla f = (\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}) = (y z, x z, x y)$ .
At the point $(0, 2, 1)$ , where $x = 0$ , $y = 2$ , $z = 1$ : $\nabla f = (2 \cdot 1, 0 \cdot 1, 0 \cdot 2) = (2, 0, 0)$ .
Normalise the direction: $∣ v ∣ = 2^{2} + 1^{2} + 2^{2} = 9 = 3$ , so $\overset{v}{^} = (\frac{2}{3}, \frac{1}{3}, \frac{2}{3})$ .
Directional derivative: $D_{\overset{v}{^}} f = (2, 0, 0) \cdot (\frac{2}{3}, \frac{1}{3}, \frac{2}{3}) = 2 \cdot \frac{2}{3} + 0 + 0 = \frac{4}{3}$
The rate of increase is $\frac{4}{3}$ , which is option (B).

KEY IDEA

The rate of increase in a direction is the directional derivative, $\nabla f \cdot \overset{v}{^}$ . You must dot the gradient with the unit vector along $v$ , not with $v$ as given.

COMMON MISTAKE

The usual slip is to dot with $v = (2, 1, 2)$ without dividing by $∣ v ∣ = 3$ . That gives $4$ , which is option (D). The other slip is to answer with $∣\nabla f ∣ = 2$ (option C), which is the largest rate of increase over all directions, not the rate along $v$ .

From the GATE 2023 EC paper. Every question of this paper · Download the full paper · All 23 Vector calculus questions, with notes

GATE 2023 EC, question 12: Vector calculus — solution and answer

Answer and worked solution

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