Differential Forms

Explore the fundamental concepts of differential forms in this comprehensive 7-hour course. Delve into the tangent space, 1-forms, and m-forms, understanding their geometric interpretations and algebraic properties. Learn about the wedge product and its significance in differential geometry. Discover how to integrate differential forms, from 1-forms to m-forms, and apply these techniques to classic problems like u-substitution and change of variables. Examine the exterior derivative, its properties, and its connection to vector calculus. Investigate the Hodge operator and its role in relating differential forms to familiar vector calculus concepts. Apply differential forms to Maxwell's equations in electromagnetism. Finally, study the integration of m-forms over m-chains and explore the boundaries of m-cells and m-chains, providing a solid foundation in this powerful mathematical framework used in physics and advanced geometry.

Differential Forms | Introduction and the Tangent Space.
Differential Forms | What is a 1-form?.
Differential Forms | The geometry of multiplying 1-forms..
Differential Forms | What is an m-form?.
Differential Forms | What is an m-form?.
Differential Forms | Algebraic Properties of the Wedge Product.
Differential Forms | The dimension of the space of m-forms..
Differential Forms | What is a differential m-form?.
Differential Forms | Integrating 2-forms.
Differential Forms | Integrating m-forms..
Differential Forms | Examples of integrating 2-forms..
Differential Forms | Classic u-substitution via integrating 1-forms..
Differential Forms | Change of variables via integrating 2-forms..
Differential Forms | The exterior derivative..
Differential Forms | The product rule for the exterior derivative..
Differential Forms | Properties of the exterior derivative..
Differential Forms | The Hodge operator..
Differential Forms | The exterior derivative and vector calculus..
Differential Forms | The Hodge operator via an inner product..
Differential Forms | The Minkowski metric and the Hodge operator..
Maxwell's equations via differential forms..
Differential Forms | Integrals of m-forms over m-chains..
Differential Forms | Boundaries of m-cells and m-chains..