GATE 2020 EE – Question 37

Engineering Mathematics · Calculus: Vector identities, Directional derivatives, Line integral, Surface integral, Volume integral, Stokes's theorem, Gauss's theorem, Divergence theorem, Green's theorem · 2 marks · Multiple choice

The vector function expressed by

$$\mathbf F=\mathbf a_x(5y-k_1z)+\mathbf a_y(3z+k_2x)+\mathbf a_z(k_3y-4x)$$

represents a conservative field, where $\mathbf a_x$, $\mathbf a_y$, $\mathbf a_z$ are unit vectors along x, y and z directions, respectively. The values of constants $k_1,k_2,k_3$ are given by:

$k_1=3,k_2=3,k_3=7$
$k_1=3,k_2=8,k_3=5$
$k_1=4,k_2=5,k_3=3$
$k_1=0,k_2=0,k_3=0$

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Correct answer: (C) $k_1=4,k_2=5,k_3=3$

Explanation

A conservative field is irrotational, so $\nabla\times\mathbf F=0$. The components are $k_3-3=0$, $-k_1+4=0$ and $k_2-5=0$, giving $k_3=3$, $k_1=4$ and $k_2=5$.