Differential Form
Introduction
A differential form is a mathematical concept used primarily in the fields of multivariable calculus, differential geometry, and algebraic topology. It is a tool that allows for the generalization of analysis on Euclidean space to more complex spaces. Differential forms are used to define integrals of multi-variable functions and have applications in many areas of physics, including electromagnetism and general relativity.
Definition
In the simplest terms, a differential form on a manifold is a smooth section of the cotangent bundle. The cotangent bundle of a manifold is the vector bundle of all the cotangent spaces at every point in the manifold. A differential form assigns to each point of the manifold a covector in the cotangent space at that point. The order of a differential form is given by the degree of the top exterior power in the tensor product decomposition of the form.
Types of Differential Forms
Differential forms can be classified into several types based on their order and the properties they possess. These include:
Zero-Forms
A zero-form on a manifold is simply a smooth function on the manifold. In other words, it is a function that assigns a real number to each point in the manifold. Zero-forms are the simplest type of differential forms.
One-Forms
A one-form on a manifold is a linear map from the tangent space at each point to the real numbers. One-forms can be thought of as infinitesimal increments in a function.
Two-Forms and Higher
Two-forms and higher are more complex and have applications in areas such as electromagnetism and general relativity. A two-form, for example, can be used to represent the electromagnetic field in a vacuum.
Operations on Differential Forms
There are several operations that can be performed on differential forms, including the exterior derivative, the wedge product, and the interior product.
Exterior Derivative
The exterior derivative is an operation that takes a differential form of order k and produces a differential form of order k+1. It is a generalization of the gradient, divergence, and curl in vector calculus.
Wedge Product
The wedge product is an operation that takes two differential forms and produces a new differential form. It is a bilinear map that is antisymmetric, meaning that the order of the forms in the product matters.
Interior Product
The interior product is an operation that takes a vector field and a differential form and produces a new differential form. It is used in the definition of the Lie derivative, which measures the change of a tensor field along the flow of a vector field.
Applications of Differential Forms
Differential forms have many applications in mathematics and physics. In mathematics, they are used in multivariable calculus, differential geometry, and algebraic topology. In physics, they are used in the formulation of Maxwell's equations of electromagnetism and Einstein's field equations of general relativity.