A differential 
-form
is a covariant tensor of tensor
rank 
that is antisymmetric under exchange of any pair
of indices. The number of algebraically
independent components in 
dimensions is given by the binomial coefficient 
. In particular, a one-form 
(often simply called a "differential")
is a quantity
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where the functions 
are the components of a covariant
tensor. Changing variables from 
to 
gives
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(2)
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(3)
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where
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which is the covariant tensor transformation law.
A 
-alternating multilinear form on a vector space 
corresponds to an element of 
, the 
th exterior power of the dual
vector space to 
. A differential 
-form on a manifold is a bundle
section of the vector bundle 
, the 
th exterior power of the cotangent
bundle. Hence, it is possible to write a 
-form in coordinates by
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where 
ranges over all increasing subsets of 
elements from 
, and the 
are functions.
An important operation on differential forms, the exterior derivative, is used in the celebrated
Stokes' theorem. The exterior
derivative 
of a 
form is a 
-form.
In fact, by definition, if 
is the coordinate function,
thought of as a zero-form, then 
.
Another important operation on forms is the wedge product, or exterior product. If 
is a 
-form and 
is a 
-form, then 
is a 
form. Also, a 
-form can be contracted
with an 
-vector,
i.e., a bundle section of 
, to give a 
-form, or if 
, an 
-vector. If the manifold has
a metric tensor, then there is an operation
dual to the wedge product, called the interior
product.
In higher dimensions, there are more kinds of differential forms. For instance, on the tangent space to 
there is the zero-form 1, two one-forms 
and 
, and one 2-form 
. A one-form can be written
uniquely as 
. In four dimensions, 
is a 2-form
that cannot be written as 
.
The minimum number of terms necessary to write a form is sometimes called the rank of the form, usually in the case of a 2-form.
When a form has rank one, it is called decomposable.
Another meaning for rank of a form is its tensor rank,
in which case a 
-form
can be described as an antisymmetric tensor
of tensor rank 
, in fact of type 
. The rank of a form can also mean the dimension
of its form envelope, in which case the rank is
an integer-valued function.
With the latter definition of rank, at each point a nonzero value of a 
-form is decomposable iff
it has rank 
.
For 
,
the zero form is also decomposable,
but its form envelope has dimension
0.
When 
is the dimension of a manifold 
, then 
is also the dimension of the
tangent space 
. Consequently, at any point where
it is nonzero, an 
-form has rank one in the minimum-number-of-terms convention.
The zero form has rank zero in this convention. For 
, a 
-form must be zero. Hence, an 
-form is called a top-dimensional
form. A top-dimensional form can be integrated without using a metric
tensor. Consequently, a 
-form can be integrated
on a 
-dimensional submanifold.
Differential forms with compact support form a
vector space with a test-form topology.
Integration over a properly embedded oriented submanifold
defines a continuous linear
functional on these test forms, hence a current.
Compact support ensures the integral
is finite; integration of arbitrary smooth forms
over a noncompact submanifold need not converge.
On an oriented manifold
with a Riemannian metric, the Hodge
star operator 
defines a map from 
-forms to 
-forms such that 
.
When 
is a smooth map, it pushes forward manifold
tangent vectors from 
to 
according to the Jacobian 
. Hence, a differential form on 
pulls back to a differential form on
.
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The pullback map is a linear transformation which commutes with the exterior derivative,
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