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OK, try this.
In ordinary [[Cartesian coordinates]], the curl is calculated as:
In ordinary [[Cartesian coordinates]], the curl is calculated as:
:<math>\nabla \cdot \vec V = \frac{\partial V_x}{\partial x} + \frac{\partial V_y}{\partial y} + \frac{\partial V_z}{\partial z}</math>
:<math>\nabla \times \vec V = (\ \ \frac{\partial V_z}{\partial y} - \frac{\partial V_y}{\partial z},\ \ \ \ \frac{\partial V_x}{\partial z} - \frac{\partial V_z}{\partial x},\ \ \ \ \frac{\partial V_y}{\partial x} - \frac{\partial V_x}{\partial y}\ \ )</math>
or, using suitable notation,
:<math>\nabla \cdot \vec V = \sum_{i=1}^3 \frac{\partial V_i}{\partial x_i}</math>
or, using a somewhat fictitious notation like the [[determinant]] notation for cross product,
:<math>\nabla \times \vec V = \begin{vmatrix}
\hat x & \hat y & \hat z \\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
V_x & V_y & V_z
\end{vmatrix}</math>
where <math>\hat x</math>, <math>\hat y</math>, and <math>\hat z</math> are the unit basis vectors.
If one thinks of <math>\nabla</math> as being a fictional vector field with components <math>(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z})</math>, one can sort of see that the cross product notation makes sense. This is also useful for remembering how to calculate a curl.
If one thinks of <math>\nabla</math> as being a fictional vector field with components <math>(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z})</math>, one can sort of see that the cross product notation makes sense. This is also useful for remembering how to calculate a curl.
The curl is a true vector field operation—the result is independent of the coordinate system that is used. The proof of that, and its ramifications, are beyond the scope of this page.
The curl is a true vector field operation—the result is independent of the coordinate system that is used. The proof of that, and its ramifications, are beyond the scope of this page.
The curl operation has an intrinsic "handedness" to it. Any physical phenomenon described by the curl operation (for example, magnetic fields), involves some kind of "right-hand rule".
The curl is an extremely important operation in physics, mathematics, and engineering. It is perhaps most famous for its appearance in [[Maxwell's Equations]].
The curl is an extremely important operation in physics, mathematics, and engineering. It is perhaps most famous for its appearance in [[Maxwell's Equations]].