GATE 2021 EC – Question 27
Consider the vector field $\mathbf F=\mathbf a_x(4y-c_1z)+\mathbf a_y(4x+2z)+\mathbf a_z(2y+z)$ in a rectangular system $(x,y,z)$ with unit vectors $\mathbf a_x,\mathbf a_y$, and $\mathbf a_z$. If the field $\mathbf F$ is irrotational (conservative), then the constant $c_1$ (in integer) is ________.
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Correct answer: 0
Explanation
For an irrotational field $\nabla\times\mathbf F=0$. The $y$-component of the curl is $\frac{\partial F_x}{\partial z}-\frac{\partial F_z}{\partial x}=-c_1-0$, which must vanish, so $c_1=0$. The other two components are already zero ($2-2=0$ and $4-4=0$).