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GATE 2024 EC – Question 54

Engineering Mathematics · Vector Analysis · 2 marks · Multiple select

Let $F_1$, $F_2$, and $F_3$ be functions of $(x,y,z)$. Suppose that for every given pair of points $A$ and $B$ in space, the line integral $\int_C(F_1dx+F_2dy+F_3dz)$ evaluates to the same value along any path $C$ that starts at $A$ and ends at $B$. Then which of the following is/are true?

  1. For every closed path $\Gamma$, we have $\oint_\Gamma(F_1dx+F_2dy+F_3dz)=0$.
  2. There exists a differentiable scalar function $f(x,y,z)$ such that $F_1=\dfrac{\partial f}{\partial x},\ F_2=\dfrac{\partial f}{\partial y},\ F_3=\dfrac{\partial f}{\partial z}$.
  3. $\dfrac{\partial F_1}{\partial x}+\dfrac{\partial F_2}{\partial y}+\dfrac{\partial F_3}{\partial z}=0$.
  4. $\dfrac{\partial F_3}{\partial y}=\dfrac{\partial F_2}{\partial z},\ \dfrac{\partial F_1}{\partial z}=\dfrac{\partial F_3}{\partial x},\ \dfrac{\partial F_2}{\partial x}=\dfrac{\partial F_1}{\partial y}$.

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Show answer and explanation

Correct answer: (A) For every closed path $\Gamma$, we have $\oint_\Gamma(F_1dx+F_2dy+F_3dz)=0$.; (B) There exists a differentiable scalar function $f(x,y,z)$ such that $F_1=\dfrac{\partial f}{\partial x},\ F_2=\dfrac{\partial f}{\partial y},\ F_3=\dfrac{\partial f}{\partial z}$.; (D) $\dfrac{\partial F_3}{\partial y}=\dfrac{\partial F_2}{\partial z},\ \dfrac{\partial F_1}{\partial z}=\dfrac{\partial F_3}{\partial x},\ \dfrac{\partial F_2}{\partial x}=\dfrac{\partial F_1}{\partial y}$.

Explanation

Path independence means the field $\vec F=(F_1,F_2,F_3)$ is conservative. Then (A) the integral around any closed loop is zero; (B) $\vec F=\nabla f$ for a scalar potential $f$; (D) $\nabla\times\vec F=0$, which is exactly the three equalities of mixed partial derivatives listed. (C) says the divergence is zero, which is not implied (e.g. $\vec F=\nabla(x^2)$ is conservative with divergence 2).