What Is the Curl of a Vector Field and Its Relation to Divergence?

[fluidistic]

Homework Statement

For example in electromagnetism and I think it's true for any vector field, the relation $$\vec \nabla \cdot (\vec \nabla \times \vec E)=0$$.
As far as I know, the curl of a vector field is a vector. So basically the above expression takes the divergence of a vector? It can't be so. It means that the curl of a vector field is not a vector but a vector field.

It sounds very strange to me... can anyone shred some light on this please?

 

[SVXX]

My 2 cents - The curl of a vector field, being a cross product, is a vector perpendicular to the field and so the dot product outside would naturally be zero, coz perpendicular vectors have a zero dot product...?

 

[fluidistic]

My 2 cents - The curl of a vector field, being a cross product, is a vector perpendicular to the field and so the dot product outside would naturally be zero, coz perpendicular vectors have a zero dot product...?

I appreciate your help, however I have some questions.
What do you mean by "dot product outside"? Dot product between what vectors?

I'm taking a divergence. Written differently I have div(curl (E))=0.
I've been taught that the curl of a vector field is a vector and I've been taught that the divergence applies to vector fields, not vectors. I'm not asking why the divergence of the curl of a vector field is worth 0, but a clarification of the divergence/curl in this special case.

 

[I like Serena]

I appreciate your help, however I have some questions.
What do you mean by "dot product outside"? Dot product between what vectors?

I'm taking a divergence. Written differently I have div(curl (E))=0.
I've been taught that the curl of a vector field is a vector and I've been taught that the divergence applies to vector fields, not vectors. I'm not asking why the divergence of the curl of a vector field is worth 0, but a clarification of the divergence/curl in this special case.

You can only take the curl of a vector field which yields a vector field.

For a vector field A this is:
[URL]http://upload.wikimedia.org/math/c/5/d/c5df8cb34c3b1480237b941f46628338.png[/URL] =
[URL]http://upload.wikimedia.org/math/a/5/9/a59969f3cc771bae6bf56bcf001aeb3d.png[/URL]
The x, y, and z with the hats on them represent the unit vectors in each direction.

If you take the divergence from this curl, this will always result in 0.
This is not trivial to think up, but if you fill everything in, you'll see that it works out.
This results in:
[URL]http://upload.wikimedia.org/math/b/8/3/b839f27612baaf13dca6770ef8f798fe.png[/URL]

 

[fluidistic]

Ok thanks a lot Serena liker. I erroneously thought that the curl of a vector field was a vector, instead of a vector field.
Now this makes perfect sense.