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GATE 2023 CH – Question 14

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

For a two-dimensional plane, the unit vectors, $(\hat{e}_r, \hat{e}_\theta)$ of the polar coordinate system and $(\hat{i}, \hat{j})$ of the cartesian coordinate system, are related by the following two equations.
$\hat{e}_r = \cos\theta\hat{i} + \sin\theta\hat{j}$
$\hat{e}_\theta = -\sin\theta\hat{i} + \cos\theta\hat{j}$
Which one of the following is the CORRECT value of $\frac{\partial(\hat{e}_r + \hat{e}_\theta)}{\partial\theta}$ ?

  1. 1
  2. $\hat{e}_\theta$
  3. $\hat{e}_r + \hat{e}_\theta$
  4. $-\hat{e}_r + \hat{e}_\theta$

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Correct answer: (D) $-\hat{e}_r + \hat{e}_\theta$

Explanation

$\frac{\partial\hat{e}_r}{\partial\theta} = -\sin\theta\hat{i} + \cos\theta\hat{j} = \hat{e}_\theta$ and $\frac{\partial\hat{e}_\theta}{\partial\theta} = -\cos\theta\hat{i} - \sin\theta\hat{j} = -\hat{e}_r$. The sum is $\hat{e}_\theta - \hat{e}_r$.