Discussion
Let I, J, and K are unit vectors along the three mutually perpendicular x, y and z axes, respectively. If \(\mathbf{F} = f\mathbf{I} + g\mathbf{J} + h\mathbf{K}\) is a continuously differentiable vector point function, then curl F is
Answer: B
The curl is the second option: \(\mathbf{I}(\frac{\partial h}{\partial y} - \frac{\partial g}{\partial z})\) \(- \mathbf{J}(\frac{\partial h}{\partial x} - \frac{\partial f}{\partial z})\) \(+ \mathbf{K}(\frac{\partial g}{\partial x} - \frac{\partial f}{\partial y})\).
The curl is the determinant
\[ \nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{I} & \mathbf{J} & \mathbf{K} \\ \partial_{x} & \partial_{y} & \partial_{z} \\ f & g & h \end{vmatrix} \]
Expanding along the first row, the I term is \(\partial_{y}h - \partial_{z}g\), the J term carries a minus sign in front of \(\partial_{x}h - \partial_{z}f\), and the K term is \(\partial_{x}g - \partial_{y}f\).
The third option has a plus sign before J, so its J component has the wrong sign. The first and fourth options reverse the order inside every bracket, which gives the negative of the curl (and the fourth also has the wrong J sign).
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