Overview

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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.

Syllabus

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..

Taught by

Michael Penn

Reviews

5.0 rating, based on 1 Class Central review

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Roger Powell

Really enjoyed this course, superbly presented using a long-forgotten device called "a chalk-board" - and how much better it was for it! I love the summary of lectures up to the current lecture at the start of each new lecture. I love the amount of…

Really enjoyed this course, superbly presented using a long-forgotten device called "a chalk-board" - and how much better it was for it! I love the summary of lectures up to the current lecture at the start of each new lecture. I love the amount of reinforcement, constantly pushing home the points explained before. I tried several examples and found I had just about got the hang of it (amazingly).

My only slightly critical warning to someone taking the course is that the number of concepts per unit time seems to increase, especially towards the end, i.e., the d^2 C/dt^2 becomes increasingly positive where C = concept.

But I started with a very poor understanding of m-forms, differential forms and the Hodge* operator and I felt I totally got it (especially the Hodge* operator, which is now completely clear to me). I had never heard of m-cells and m-chains but it wasn't too difficult to understand. The Stokes' theorem proof was really great!

Long may chalk-boards remain in use!

Taught by

Differential Forms - The Hodge Operator via an Inner Product

Differential Forms - The Minkowski Metric and the Hodge Operator

Differential Forms - The Exterior Derivative and Vector Calculus

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