# 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.

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. $$\nabla\cdot\nabla\times V=0$$.

• In terms of differenetial operators what would del cross vs dot be like? Apr 17, 2022 at 21:22
• @Minikute $\nabla\times$ needs to take a vector, which $\nabla\cdot$ can't return. But a third kind of del, the grad of a scalar, $(\nabla\phi)_i:=\partial_i\phi$, is a vector you can pass to curl, viz. $\nabla\times\nabla\phi=0$ (again, this is what you'd expect, because the derivatives commute). Edit: I just noticed your "vs", but I'll leave what I already wrote. Could you clarified what you meant by curl vs div?
– J.G.
Apr 17, 2022 at 21:26
• i don't know vector calculus Apr 17, 2022 at 21:39
• I want the formula in partial different form, this the expanded version of that kinky notation Apr 17, 2022 at 21:42
• I've given you that. For example, $(\nabla\times V)_1=\partial_2V_3-\partial_3V_2$.
– J.G.
Apr 17, 2022 at 21:45

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.

• So how is it different is what i asked ofc it's a notation but that's not my question Apr 17, 2022 at 21:19
• Without notation what's the expanded form in partial differential operators. Apr 17, 2022 at 21:25
• Look up the definitions of curl and divergence, that will give more than any answer here could. instead of actually computing a cross product or dot product, where you would multiply the "vectors", you apply the differential operator Apr 17, 2022 at 21:33
• How is this tied to vectors Apr 17, 2022 at 21:38
• Briefly: The curl is a measure of the rotation of a field at a point. And the divergence is a measure of how much the vector field flows out of a point. For the intuition behind the curl and divergence, look up "divergence theorem" or "stokes theorem" Apr 17, 2022 at 21:39