Maxwell's eq-meaning of del's cross and dot product? In maxwell's eq there is del whose cross and dot products exist.
So what is del in cross vs dot product.
What's the difference when it's just a partial differential operator.
 A: You're used to the definitions$$U\cdot V=U_iV_i,\,(U\times V)_i:=\epsilon_{ijk}U_jV_k.$$(I've used Einstein notation without worrying about index heights.) Similarly,$$\nabla\cdot V=\partial_iV_i,\,(\nabla\times V)_i:=\epsilon_{ijk}\partial_jV_k.$$Since derivatives don't commute with functions, the consequences are slightly more complicated than for "normal" vectors. For example, in$$\nabla\times(U\times V)=U(\nabla\cdot V)-V(\nabla\cdot U)+\color{red}{(V\cdot\nabla)U-(U\cdot\nabla)V},$$the red terms have no "vanilla" analog. On the other hand, derivatives commute with each other, so e.g. $\nabla\cdot\nabla\times V=0$.
A: They aren't real cross/dot products, it is a notational trick.
$\nabla × $ is called the curl.
$\nabla \cdot$ is called the divergence.
$\nabla \cdot \vec{F} = \frac{\partial F_{x}}{\partial x} \hat i + \frac{\partial F_{y}}{\partial y} \hat j + \frac{\partial F_{z}}{\partial z} \hat k$
$\nabla × \vec{F} = (\frac{\partial F_{z}}{\partial y} - \frac{\partial F_{y}}{\partial z})\hat i +(\frac{\partial F_{x}}{\partial z} - \frac{\partial F_{z}}{\partial x})\hat j +(\frac{\partial F_{y}}{\partial x} - \frac{\partial F_{x}}{\partial y})\hat k $
Curl and divergences computation is the same as a cross product or dot product, but instead  multiplying the vectors, you differentiate the component.
