Divergence and curl rule simplification?

[Durato]

Just for reference, i got this question from reading an online ebook:
http://www.plasma.uu.se/CED/Book/EMFT_Book.pdf
The bottom equation on page 24 is where i these equations came up.

I have been reading some stuff and i keep coming across an annotation which looks exactly
like a divergence symbol except backwards.
V dot (del sign)

Specifically, it was used as follows:
[j(x') dot del']del'(1/(abs(x-x')))

where del is the upside triangle symbol and x and x' are just two different variables.
del' operates only on the x' symbol.

Also, can anyone point me out to any online reference which shows how to simplify
vector identities which include integrals. Example:

Homework Statement

Simplify Kcurl(curl(integral(d^3x'*j(x')/abs(x-x'))))
where, again, x and x' are two independent variables and d^3x' just represent the volume
differential (not a variable!)

Homework Equations

curl(curl(a)) = grad(div(a))-laplace(a)

The Attempt at a Solution

No clue where to start. But their solution was as follows:
= -K*(integral(d^3x'j(x')laplace(1/(abs(x-x'))))
+K*(integral(d^3x'[j(x') dot del']grad'(1/(abs(x-x')))))

where del' or grad' just means do in terms of x' variable.

I got lost when they started using grad' and dot del' in the second integral, even though
they had used a curl in terms of x, not x'.

 

[e(ho0n3]

Read this page: http://en.wikipedia.org/wiki/Del

That should take care of your first concern. Apropos your second concern, the integral seems to be acting like a scalar.