The definition of Differential Forms
A differential 1 form on a manifold is a sooth map
of the tangent bundle of to the line, linear on each tangent space
One could say that a differential 1-form on is an algebraic 1-form on which is “differentiable with respect to .”
Also we can define the differential forms , which act on a tangent vector will produce its components , that is
Every differential 1-form on the space with a given coordinate system can be written uniquely in the form
where the coefficients are smooth functions.
The usual differential of a function at is in fact a differential 1-form.
Notice that itself will not give a value on , it should act on a vector in the tangent space and then give the value.
For example, let be the velocity vector of the curve . Then, the value of differential 1-form on at point is
A differential k-form at a point of a manifold is an exterior k-form on the tangent space to at , i.e., a k-linear skew-symmetric function of vectors tangent to at .
Every differential k-form on the space with a given coordinate system can be written uniquely in the form
where the are smooth functions on .
Integration of Differential Forms
be a differential form and S a differentiable k-manifold over which we wish to integrate, where S has the parameterization
for in the parameter domain . Then (Rudin 1976) defines the integral of the differential form over as
where the integral on the right-hand side is the standard Riemann (or Lebesgue) integral over D, and
is the determinant of the Jacobian. The Jacobian exists because is differentiable.
Immediately, we can see that the integral of a k-form on euclidean space over a domain is
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree.
If a k-form is thought of as measuring the flux through an infinitesimal k-parallelotope, then its exterior derivative can be thought of as measuring the net flux through the boundary of a (k+1)-parallelotope.
The exterior derivative of a differential form of degree k is a differential form of degree k+1.
In a local coordinate system on , the form is written as
then the exterior derivative is
 Arnol’d, Vladimir Igorevich. Mathematical methods of classical mechanics. Vol. 60. Springer Science \& Business Media, 2013.