The set of differential
-forms on a manifold
(example
) is a vector space
and when
then
is the set of smooth functions. Thus smooth functions are 0-forms. Now
-forms are integrated on
-manifolds. For example, a 1-form
can be integrated on a curve
. Also differential forms can be differentiated using the operator d called the exterior operator where
acts on a
-form to produce a
-form and that
.
Now
1. if
, then
is a 1-form so that
. Thus
is the gradient operator
.
2. If we have a 1-form
, then
which is a two form. In this case we have
is the
operator.
3. If we have 2-form
then again get a 3-form
. If we think of
as a function
, then again we get
is the
operator.