Differential Forms, Equations, and Electromagnetism

Differential forms, differential equations, and their relationship are pivotal in the construction of the theory of electromagnetism. Recently, I’ve been very intrigued by this exact thing, and tried tying everything together in the context of electrostatics.

These notes are on my recent strides in these subjects, also available on notes. I’m particular proud of the second document, where I tightly related these topics in a way I thought was appropriate. Hope it helps anyone who comes across it!

These notes were for learning the basics of differential forms!

Download it here.

These notes were for the midterms of electromagnetism 1 (PHY2009-01)!

Download it here.

I record here a few note I’d like to make. Throughout the journey, I was able to realise the importance of the field of real analysis with it providing a vigorous foundation for integration in higher dimensions and on manifolds. Albeit that I’ve understood only a small portion of the whole, I was also able to sense that I wanted to further investigate what it meant to integrate over singularities like those provided by point charges.

I’d like to mention that upon studying electrostatics, I was again suprised by the power of abstraction and generalisation. There was a huge chunk of detail in veil when electrostatics was explained by Griffiths’ book on Electrodyanamics. Frankly, I found the book, along with the book by Jackson, extremely chaotic and baffling in its way of explaining things (I’d go a whole day talking about it). I’m certain that I would’ve found the whole subject impossible to tackle if not for the separate studying. In this sense, I want to take this opportunity to emphasize to others and my future self the following: when things seem incomprehensible, the fault is often on how, the way that you’re going about it, rather than how much you’re able to tackle the ideas.