Geometry Differential Geometry: Book on its applications?

AI Thread Summary
The discussion centers on finding applications of differential forms and differential geometry beyond relativity. The original poster expresses a desire for resources that explore practical applications, particularly in areas like aerodynamics, and mentions an interest in fractional calculus. Several book recommendations are provided, including Schutz's work on applications in physics, Burke's "Applied Differential Geometry," and resources on discrete differential geometry. The conversation also touches on geometric mechanics, control theory, and information geometry, with participants sharing links to various texts and papers. There is a focus on the intersection of differential geometry with advanced mechanics and stochastic geometry, highlighting the need for further exploration of these topics in practical contexts. The discussion concludes with mentions of specific mathematical concepts like sheaves and cohomology, indicating a deeper dive into the theoretical aspects of differential geometry.
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Hi, I'm already familiar with differential forms and differential geometry ( I used multiple books on differential geometry and I love the dover book that is written by Guggenheimer. Also used one by an Ian Thorpe), and was wondering if anyone knew a good book on it's applications. Preferably not just in the realm of relativity. I used to study a lot of pure mathematics topics and now I'm leaning towards applications and I've noticed that there is very little on the subject that I can find, which I think is a shame because it is my favorite subject (Fractional calculus is cool too, I even found some papers on fractional differential forms and geometry). Any suggestions? I'm flexible on this; it doesn't have to be a dedicated differential geometry book but I'd say if it has a few good chapters on applications that would be neat. I'm not too familiar on the subject of aerodynamics, does anyone know if that subject uses it? Thanks for your help! -James
 
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Looks very promising, thank you!
 
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Possibly useful [...these are on my to do list... someday]:

Hirani, Anil Nirmal (2003) Discrete exterior calculus. Dissertation (Ph.D.), California Institute of Technology. https://resolver.caltech.edu/CaltechETD:etd-05202003-095403

Crane, Keenan. Discrete Differential Geometry: An Applied Introduction
https://www.cs.cmu.edu/~kmcrane/Projects/DDG/
https://www.cs.cmu.edu/~kmcrane/Projects/DDG/paper.pdf

Bossavit, Alain. Computational Electromagnetism: Variational Formulations, Complementarity, Edge Elements
https://www.amazon.com/gp/product/0123885604/?tag=pfamazon01-20
related:
https://www.researchgate.net/publication/254470625_On_the_geometry_of_electromagnetism
https://www.researchgate.net/publication/200018385_Differential_Geometry_for_the_student_of_numerical_methods_in_Electromagnetism
https://www.researchgate.net/publication/242462763_Computational_electromagnetism_and_geometry_Building_a_finite-dimensional_Maxwell's_house
Daverz said:
Burke's Applied Differential Geometry

https://www.amazon.com/dp/0521269296/?tag=pfamazon01-20
The errata for the Burke's book is at
http://www.ucolick.org/~burke/forms/errata.ps
linked from http://www.ucolick.org/~burke/class/adg.html
 
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Thanks for the additional references. I've started the homological algebra and have previously read some of Kranes paper I like them alot. I haven't seen that before. I had a different book on discrete differential geometry but it was very jumbled with different topics kind of piled one on top of each other with no seeming attention to its order. I find the discrete stuff very interesting. I wonder if there is a discrete analog of hyperbolic geometry or if you can construct such a thing using Krane's stuff? I'll definitely read that thank you.
 
I do not know if this is still of interest to you

Differential geometry applied to classical mechanics is sometimes known as geometric mechanics.
https://www.amazon.com/dp/0387968903/?tag=pfamazon01-20
https://www.amazon.com/dp/038798643X/?tag=pfamazon01-20
https://www.amazon.com/dp/0821844385/?tag=pfamazon01-20
https://www.amazon.com/dp/0387967907/?tag=pfamazon01-20
https://www.amazon.com/dp/0521636361/?tag=pfamazon01-20
https://www.amazon.com/dp/0199212902/?tag=pfamazon01-20
https://www.amazon.com/dp/184816775X/?tag=pfamazon01-20
https://www.amazon.com/dp/1848167784/?tag=pfamazon01-20
https://www.amazon.com/dp/3319569511/?tag=pfamazon01-20

Control Theory
https://www.amazon.com/dp/1493930168/?tag=pfamazon01-20

Solid Mechanics
https://www.amazon.com/dp/9814616036/?tag=pfamazon01-20

Differential geometry applied to information theory is known as information geometry.
https://www.amazon.com/dp/B01BDF0CQM/?tag=pfamazon01-20
https://math.ucr.edu/home/baez/information/

Bruce J. West has written several books on fractional calculus. I know that he cares about applications.
 
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caz said:
I do not know if this is still of interest to you

Differential geometry applied to classical mechanics is sometimes known as geometric mechanics.
https://www.amazon.com/dp/0387968903/?tag=pfamazon01-20
https://www.amazon.com/dp/038798643X/?tag=pfamazon01-20
https://www.amazon.com/dp/0821844385/?tag=pfamazon01-20
https://www.amazon.com/dp/0387967907/?tag=pfamazon01-20
https://www.amazon.com/dp/0521636361/?tag=pfamazon01-20
https://www.amazon.com/dp/0199212902/?tag=pfamazon01-20
https://www.amazon.com/dp/184816775X/?tag=pfamazon01-20
https://www.amazon.com/dp/1848167784/?tag=pfamazon01-20
https://www.amazon.com/dp/3319569511/?tag=pfamazon01-20

Control Theory
https://www.amazon.com/dp/1493930168/?tag=pfamazon01-20

Solid Mechanics
https://www.amazon.com/dp/9814616036/?tag=pfamazon01-20

Differential geometry applied to information theory is known as information geometry.
https://www.amazon.com/dp/B01BDF0CQM/?tag=pfamazon01-20
https://math.ucr.edu/home/baez/information/

Bruce J. West has written several books on fractional calculus. I know that he cares about applications.

Yes I am still interested, which one would apply the most of geometric arguments in your opinion(in regard to the dynamics books)? I've seen stuff on information geometry, personally I prefer stochastic geometry over that. I am a little biased though because I have more experience with topics relating to the latter though.
 
s00mb said:
Yes I am still interested, which one would apply the most of geometric arguments in your opinion(in regard to the dynamics books)? I've seen stuff on information geometry, personally I prefer stochastic geometry over that. I am a little biased though because I have more experience with topics relating to the latter though.

I’ve gotten interested in advanced mechanics over COVID, so you are seeing a list of interesting things that I have found. I apologize for not having read and absorbed them all so that I can give a good review 😜
 
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  • #10
caz said:
I’ve gotten interested in advanced mechanics over COVID, so you are seeing a list of interesting things that I have found. I apologize for not having read and absorbed them all so that I can give a good review 😜
That's no problem, the one about dynamics on manifolds seems to be the one for me to look at. I like manifold theory too, I imagine they'd apply some tensors or Riemann geometry in it.
 
  • #12
Check out "Global Calculus" by S. Ramanan. It is not about applications, but contains material/approach that is not generally discussed in books on differential geometry.
 
  • #13
love_42 said:
contains material/approach that is not generally discussed in books on differential geometry
Such as?
 
  • #14
Demystifier said:
Such as?
Sheaves, exact sequences, cohomology.
 
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