GATE 2026 CH – Question 21

Engineering Mathematics · Calculus: Gradient, divergence, curl, vector identities and integral theorems · 1 mark · Multiple choice

A two-dimensional temperature profile is given as $T = 2x^2 + 3xy + y^2$. If $\hat{i}$ and $\hat{j}$ are the unit vectors along x and y directions, respectively, which one of the following is the directional derivative of T at the location $x = 2$, $y = 2$?

$8\hat{i} + 12\hat{j}$
$14\hat{i} + 10\hat{j}$
$8\hat{i} - 12\hat{j}$
$14\hat{i} - 10\hat{j}$

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Correct answer: (B) $14\hat{i} + 10\hat{j}$

Explanation

The gradient is $\nabla T = (4x + 3y)\hat{i} + (3x + 2y)\hat{j}$. At $(2, 2)$ this is $(8 + 6)\hat{i} + (6 + 4)\hat{j} = 14\hat{i} + 10\hat{j}$, which is the option given as the directional derivative.