Relationship of curl and cross product.

[10Exahertz]

Hi all, I am very confused on how to define the vector product or cross product in a physical sense. I know the vector product is a psuedovector, and that it is the area of a parallelogram geometrically. However, I know it used used to describe rotation in physics. As with torque, magnetism and angular momentum. I was wondering how the curl operator plays a role in determining the vectors direction in a cross product or vector product, and why its magnitude is its parallelograms area. I understand the curl operator, but for some reason cannot directly connect it to a cross product to understand it.
much thanks

 

[Khashishi]

In modern notation, the curl of A is written $\nabla \times \mathbf{A}$.
It is the cross product of the "del" operator $\nabla$ with the vector. The "del" operator is basically shorthand for
$\nabla = \frac{\partial}{\partial x}\hat{e}_x + \frac{\partial}{\partial y}\hat{e}_y + \frac{\partial}{\partial z}\hat{e}_z$

You should verify when you calculate the cross product between "del" (treated like a vector) and A, you get curl A. (Del is actually a vector operator.)