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SSC Pure Mathematics QUESTION #1269
Question 1
The Cauchy-Riemann equations in polar form are:
  • \(\dfrac{\partial u}{\partial r}=\dfrac{\partial v}{\partial\theta},\quad \dfrac{\partial u}{\partial\theta}=-\dfrac{\partial v}{\partial r}\)
  • \(\dfrac{\partial u}{\partial r}=\dfrac{1}{r}\dfrac{\partial v}{\partial\theta},\quad \dfrac{\partial u}{\partial\theta}=-r\dfrac{\partial v}{\partial r}\)✔️
  • \(r\dfrac{\partial u}{\partial r}=\dfrac{\partial v}{\partial\theta},\quad \dfrac{\partial u}{\partial\theta}=-r\dfrac{\partial v}{\partial r}\)
  • \(\dfrac{\partial u}{\partial r}=\dfrac{\partial v}{\partial\theta},\quad r\dfrac{\partial u}{\partial\theta}=\dfrac{\partial v}{\partial r}\)
Correct Answer Explanation
In polar coordinates \(z=re^{i\theta}\), the Cauchy-Riemann equations become: \(\dfrac{\partial u}{\partial r}=\dfrac{1}{r}\dfrac{\partial v}{\partial\theta}\) and \(\dfrac{1}{r}\dfrac{\partial u}{\partial\theta}=-\dfrac{\partial v}{\partial r}\). These are the standard polar C-R equations.