Resources

On this page, I list and review materials I have come across, enjoyed, and learned from in the areas of mathematics and physics that interest me.

Lectures

Differential forms by Michael Penn

The serie Differential Forms by Michael Penn are comprehensive lectures covering the exterior product, the exterior derivative, and the Hodge operator. The Hodge dual in 4-dimensional Minkowski space is included and Maxwell’s equations via differential forms introduced.

Maxwell’s theory in relativistic notations by Neil Turok

In 14/15 PSI - Relativity - Lecture 1, Neil Turok gives a concise and precise presentation of Maxwell’s equations in tensor notation.

Representations of Maxwell’s equations by Peter Joot

In his video series, Representations of Maxwell’s equations, Peter Joot, explores a wide range of formulations of Maxwell’s equations in great detail:

• Vector formulation

• Tensor formulation

• Geometric algebra formulation

• Spacetime algebra formulation

• Differential forms formulation

I appreciated the thorough step-by-step derivations. For completeness, since the forumlation in terms of quaternions is not covered in these videos, Peter Joot has nontheless a blog post about this. I have watched the whole video series but have not gone through the details of the blog post.

Review of manifold basics by Ruth Gregory

In her Gravitational Physics Lecture of January 8, 2024, Ruth Gregory gives a rushed overview of Manifold basics. This is not learning material though, but review material.

Cartan’s method of moving frames, tetrads, and Cartan’s structure equations

This covers the tetrad formalism, non-coordinate bases, Cartan’s method of moving frames, and Cartan’s structure equations. The more interesting the topic, the fewer resources you’ll find online. This is my playlist. I recommend starting slowly with James Cook:

• General Relativity: Lecture 20: tetrad method, Lorentzian frames, 5-16-23

• General Relativity: Lecture 20 part2: calculating via tetrad formalism, future reading

In her lecture on Cartan’s formalism, Ruth Gregory relates the covariant derivative to the abstract definition of a derivation. She presents Cartan’s method of moving frames and derives Cartan’s first structure equation. This is not for the faint-hearted. Be familiar with the concept of a non-coordinate basis. Fasten your seatbelt and be prepare to watch thrice:

• PSI 2018/2019 - Gravitational Physics - Lecture 3 - Cartan’s formalism: connection & curvature

Gravitational physics lecture by Ruth Gregory

There is great content from the Perimeter Institute Recorded Seminar Archive that I am going through. Here is a list of videos I intend to watch later.

• Gravitational Physics Lecture 1

• Gravitational Physics Lecture 2

• Gravitational Physics Lecture 3

• Gravitational Physics Lecture 4

• PSI 2017/2018 - Gravitational Physics - Lecture 3: Applying Cartan: Spherically Symmetric Spacetimes – Schwarzschild, Relativistic Stars.

Relativity and cosmology II

I am yet to watch the serie Relativity and Cosmology II by Claus Kiefer.

Cartan

I am interested in Cartan connections and Cartan structure equations and the following content:

• 10.1 - 3. Vectors, covectors and tensors.

• 10.4 - 5. The exterior algebra and exterior 𝑝-forms.

• 10.6. Connections and Cartan equations.

• 10.7 - 8. Metricity, applications, integration, Maxwell and Einstein equations.

• Connections, Curvature and Covariant Derivatives by Taylor Dupuy

Q&A

I sometimes but rarely participate in discussions, which are gathered here:

Physics Stackexchange

• Deriving the Electromagnetic Tensor