EC > GATE 2024 > Vector Calculus
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 $∫_C (F_1 dx + F_2 dy + F_3 dz)$ 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 $Γ$, we have $∫_Γ (F_1 dx + F_2 dy + F_3 dz) = 0$.
B
There exists a differentiable scalar function $f(x, y, z)$ such that $F_1 = \frac{∂f}{∂x}$, $F_2 = \frac{∂f}{∂y}$, $F_3 = \frac{∂f}{∂z}$
C
$\frac{∂F_1}{∂x} + \frac{∂F_2}{∂y} + \frac{∂F_3}{∂z} = 0$.
D
$\frac{∂F_3}{∂y} = \frac{∂F_2}{∂z}$, $\frac{∂F_1}{∂z} = \frac{∂F_3}{∂x}$, $\frac{∂F_2}{∂x} = \frac{∂F_1}{∂y}$

Correct : A;B;D

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