The differential of a smooth function is in a sense container which conceals all directional derivatives along all directions, and dependence on is linear.
If we consider the directional Lie derivative for a form of degree , then simple computations show that is no longer equal to . However, one can “correct” the Lie derivative in such a way that the result will depend on linearly. For instance, if and is a vector field, we can define the form by the identity and show that the 2-form is indeed bilinear antisymmetric.
The 2-form is called the exterior derivative of and denoted . The correspondence is an -linear operator which satisfies the Leibniz rule and for any function .
It turns out that this exterior derivation can be extended to all -forms preserving the above properties and is a nice (algebraically) derivation of the graded exterior algebra .
The lecture notes are available here.