dot or
inner or scalar product of two vectors:

a . b =
|a| |b| cos(theta)

If the dot product of nonzero vectors are zero, then the vectors are
perpendicular

cross or
outer or vector product of two vectors:

a x b = |a|
|b| sin(theta)u ; where u is unit vector normal to the
plane of ab

this is a
vector whose magnitude is the area of the parallelogram formed by a & b

math images from

http://hyperphysics.phy-astr.gsu.edu/hbase/vecal.html

gradient of
f

Div of E

Curl of E

**vector**** potential** is a vector field whose curl is a given
vector field.

Formally, given a vector field **v**, a *vector
potential* is a vector field **E** such that

** V** =

This is analogous to a *scalar potential*, which is
a scalar field whose negative gradient is a given vector field

Given a vector field **F**, its scalar potential *f *is
a scalar field whose negative gradient is **F**

**F =**** - **

Laplace
operator on f = 0 is called

In physics and mathematics, in the area of
vector calculus, **Helmholtz's**** theorem**,
also known as the **fundamental theorem of vector calculus**, states that
any sufficiently smooth, rapidly decaying vector field in three dimensions can
be resolved into the sum of an irrotational
(curl-free) vector field and a solenoidal
(divergence-free) vector field; this is known as the **Helmholtz****
decomposition**. It is named for Hermann von Helmholtz.

This implies that any such vector field **F**
can be considered to be generated by a pair of potentials: a scalar potential
φ and a vector potential **A**. (???)

http://en.wikipedia.org/wiki/Helmholtz_decomposition