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GATE 2025 CH – Question 36

Engineering Mathematics · Calculus: Gradient, divergence, curl, vector identities and integral theorems · 2 marks · Multiple choice

Consider a Cartesian coordinate system with orthogonal unit basis vectors $\hat{i}$, $\hat{j}$ defined over a domain: $x, y \in [0, 1]$. Choose the condition for which the divergence of the vector field $\mathbf{v} = ax\hat{i} - by\hat{j}$ is zero.

  1. $a - b = 0$
  2. $a < b$
  3. $a > b$
  4. $a + b = 0$

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Correct answer: (A) $a - b = 0$

Explanation

The divergence is $\nabla \cdot \mathbf{v} = \frac{\partial(ax)}{\partial x} + \frac{\partial(-by)}{\partial y} = a - b$. It is zero when $a - b = 0$.