Overview

This course on Differential Forms aims to teach students the concepts and applications of differential forms. By the end of the course, learners will be able to understand the geometry and algebraic properties of differential forms, perform integration of different forms, apply the exterior derivative and Hodge operator, and explore applications in vector calculus and Maxwell's equations. The course uses a combination of theoretical explanations, examples, and practical exercises to help learners grasp the material. This course is intended for individuals interested in advanced mathematics, particularly those studying topics like calculus, linear algebra, and physics.

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!

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