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I’m looking for a beautiful and unifying formalism (which I saw one of professors using and i cant ask him bc he doesn't like me at all ) that elegantly explains and derives concepts like:

  • Vectors, inner and outer products,
  • All del operations (gradient, divergence, curl, Laplacian, etc.),
  • The derivation of metrics and the geometry of the space we’re working in,
  • All of this without relying heavily on matrix or vector representations.

somthing that can handle these concepts in any coordinate system and provides a clear understanding of the underlying mathematics. For example, I’d like to see how the tedious and cumbersome del operator expressions (like the gradient or Laplacian) are derived in a way that reveals the "shape" of the space we’re working in.

I have no idea what this formalism is called or where to find it. Could anyone recommend books, courses, or resources that cover this topic in a clear and rigorous way?

Thanks in advance for your help!

Duong Ngo
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Zahra
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    There are simply too many! Have you at least searched for some? – naturallyInconsistent Feb 10 '25 at 06:59
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    This question is about math, not physics. – Ghoster Feb 10 '25 at 07:00
  • derived in a way that reveals the "shape" of the space we’re working in It sounds like you might be talking about differential geometry or tensor calculus. – Ghoster Feb 10 '25 at 07:03
  • Are you sure you mean the shape of the space vs. the shape of various coordinate systems in good old Euclidean space? For example, do you want to know how to find the Laplacian in spherical coordinates? – Ghoster Feb 10 '25 at 07:07
  • Are you searching for the de Rham complex? https://en.wikipedia.org/wiki/De_Rham_cohomology – Jeanbaptiste Roux Feb 10 '25 at 07:12
  • What course was the professor teaching when you saw this formalism? – Ghoster Feb 10 '25 at 07:17
  • @Ghoster oh yes! It is differential geometry and i think topology in general! So is there a good source to start from 0? – Zahra Feb 10 '25 at 07:56
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    The technology you are looking for is differential forms. It's usually taught in a course on differential geometry. – Mozibur Ullah Feb 10 '25 at 08:08
  • Based on your list you don't seem to be looking for a reference on vector spaces, but instead on affine spaces or more generally smooth manifolds. – Filippo Feb 10 '25 at 08:22
  • @MoziburUllah i was doing some research and yes it is differential geometry! Thanks! – Zahra Feb 10 '25 at 08:36

2 Answers2

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I think you are looking for references to Clifford's algebras and Hestenes' geometric calculus. A good starting point could be Taylor's textbook "An Introduction to Geometric Algebra and Geometric Calculus" (2021). It's a book accessible to anyone with a reasonable preparation in linear algebra and multivariable analysis.

As a reference, there is the classical textbook by Hestenes and Sobczyk "Clifford algebra to geometric calculus: a unified language for mathematics and physics" (Vol. 5). Springer Science & Business Media (2012).

See this post about the equivalence with differential forms on manifolds.

GiorgioP-DoomsdayClockIsAt-89
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I’m looking for a beautiful and unifying formalism ...

This is the theory of differential forms on manifolds. An excellent introduction to this is the book by Lee, Smooth Manifolds. If you have come across tensor fields, say in a course on General Relativity, then differential forms are basically alternating covariant tensor fields. It's a different language for the same thing.

that elegantly explains and derives concepts like ... all del operations (gradient, divergence, curl, Laplacian, etc).

To do this precisely requires a metric on the manifold. This is explained in Lee, Riemannian Manifolds. One aspect of this, is that if one forgets about turning covectors into vectors and vice-versa - which is where the metric is used - and focuses solely on covectors, then the natural operator to consider is simply the exterior derivative which transforms a differential form into a closed differential form of one degree higher. Thus in this language grad, curl & div are encoded by the exterior derivative. This applies to any dimension of space. You are not restricted to 3d. In this language the four equations of electromagnetism written in the language of vector analysis turns into two equations of differential forms and this is valid for any dimension of space and for any curved space endowed with a metric.

...are derived in a way that reveals the "shape" of the space we’re working in.

The cohomology of the de Rham complex of differential forms is equivalent to the singular cohomology of the space. Thus it is a topological invariant. This is one way to describe the 'shape' of space.

somthing that can handle these concepts in any coordinate system and provides a clear understanding of the underlying mathematics.

This is the reasoning behind differential geometry on manifolds. The idea is to explore how calculus can be made synthetic (does not rely on coordinates) rather than analytic (does rely on coordinates). The route begins with manifolds which is the coordinate free description of curved space and then differential forms encodes coordinate-free calculus on manifolds.

Another technology is Geometric Algebra. This lifts differential forms to calculus with Clifford algebras. In this language, the four equations of electromagnetism reduces to one equation of geometric algebra.

Mozibur Ullah
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