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SSC Pure Mathematics QUESTION #11143
Question 1
The Cauchy-Riemann equations in polar form are:
  • $\dfrac{\partial u}{\partial r} = \dfrac{1}{r}\dfrac{\partial v}{\partial \theta},\quad \dfrac{\partial v}{\partial r} = \dfrac{-1}{r}\dfrac{\partial u}{\partial \theta}$✔️
  • $\dfrac{\partial u}{\partial r} = \dfrac{-1}{r}\dfrac{\partial v}{\partial \theta},\quad \dfrac{\partial v}{\partial \theta} = \dfrac{-1}{r}\dfrac{\partial u}{\partial \theta}$
  • $\dfrac{\partial u}{\partial r} = \dfrac{-1}{r}\dfrac{\partial v}{\partial \theta},\quad \dfrac{\partial v}{\partial r} = \dfrac{-1}{r}\dfrac{\partial u}{\partial \theta}$
  • $\dfrac{\partial u}{\partial r} = \dfrac{1}{r}\dfrac{\partial v}{\partial \theta},\quad \dfrac{\partial v}{\partial r} = \dfrac{1}{r}\dfrac{\partial u}{\partial \theta}$
Correct Answer Explanation
The Cauchy-Riemann equations in polar coordinates $(r,\theta)$ are: $\dfrac{\partial u}{\partial r} = \dfrac{1}{r}\dfrac{\partial v}{\partial\theta}$ and $\dfrac{\partial v}{\partial r} = -\dfrac{1}{r}\dfrac{\partial u}{\partial\theta}$. These are derived by substituting $x = r\cos\theta$, $y = r\sin\theta$ into the Cartesian C-R equations $u_x = v_y$, $u_y = -v_x$.