Editing Del (section)

===Curl===
The [[Curl (mathematics)|curl]] of a vector field <math>\mathbf v(x, y, z) = v_x\hat\mathbf x  + v_y\hat\mathbf y + v_z\hat\mathbf z</math> is a [[vector field|vector]] function that can be represented as:
:<math>\operatorname{curl}\mathbf v = \left({\partial v_z \over \partial y} - {\partial v_y \over \partial z} \right) \hat\mathbf x + \left({\partial v_x \over \partial z} - {\partial v_z \over \partial x} \right) \hat\mathbf y + \left({\partial v_y \over \partial x} - {\partial v_x \over \partial y} \right) \hat\mathbf z = \nabla \times \mathbf v</math>

The curl at a point is proportional to the on-axis torque that a tiny pinwheel would be subjected to if it were centered at that point.

The vector product operation can be visualized as a pseudo-[[determinant]]:
:<math>\nabla \times \mathbf v = \left|\begin{matrix} \hat\mathbf x & \hat\mathbf y & \hat\mathbf z \\[2pt] {\frac{\partial}{\partial x}} & {\frac{\partial}{\partial y}} & {\frac{\partial}{\partial z}} \\[2pt] v_x & v_y & v_z \end{matrix}\right|</math>

Again the power of the notation is shown by the product rule:
:<math>\nabla \times (f \mathbf v) = (\nabla f) \times \mathbf v + f (\nabla \times \mathbf v)</math>

The rule for the vector product does not turn out to be simple:
:<math>\nabla \times (\mathbf u \times \mathbf v) = \mathbf u \, (\nabla \cdot \mathbf v) - \mathbf v \, (\nabla \cdot \mathbf u) + (\mathbf v \cdot \nabla) \, \mathbf u - (\mathbf u \cdot \nabla) \, \mathbf v</math>