GATE ECE 2024 | Question: 44

Question

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 $\int_{C}\left(F_{1} \mathrm{~d} x+F_{2} \mathrm{~d} y+F_{3} \mathrm{~d} z\right)$ 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?

• A. For every closed path $\Gamma$, we have $\oint_{\Gamma}\left(F_{1} \mathrm{~d} x+F_{2} \mathrm{~d} y+F_{3} \mathrm{~d} z\right)=0$.
• B. There exists a differentiable scalar function $f(x, y, z)$ such that $F_{1}=\frac{\partial f}{\partial x}, F_{2}=\frac{\partial f}{\partial y}, F_{3}=\frac{\partial f}{\partial z}$.
• C. $\frac{\partial \text{F}_{1}}{\partial x}+\frac{\partial \text{F}_{2}}{\partial y}+\frac{\partial \text{F}_{3}}{\partial z}=0$
• D. $\frac{\partial \text{F}_{3}}{\partial y}=\frac{\partial \text{F}_{2}}{\partial z}, \frac{\partial \text{F}_{1}}{\partial z}=\frac{\partial \text{F}_{3}}{\partial x}, \frac{\partial \text{F}_{2}}{\partial x}=\frac{\partial \text{F}_{1}}{\partial y}$