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Importance of differential geometry in physics? 
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#1
Jan1613, 08:03 PM

P: 162

How important is differential geometry in physics? Can someone give me some applicable fields?



#2
Jan1713, 01:22 AM

P: 530

General relativity and finding the shortest distance to Grandma's house over the hills and far away.



#3
Jan1713, 03:34 AM

P: 43

Are you referring to Geodesics (shortest path between two points)? 


#4
Jan1713, 03:49 AM

P: 530

Importance of differential geometry in physics?



#5
Jan1713, 04:10 AM

P: 43

It is used to derive Einstein's field equations to describe the curvature of Spacetime in the presence of a body of mass and energy. This curvature of Spacetime results to the phenomenon we know as gravity, including how the planets move in orbit around the Sun, among others. The mathematics (Differental Geometry) used in Einstein's field equations also predicted the existence of black holes, including concepts of the Event Horizon and Singularity as key and important regions in black holes. (I think it was the Schwarzschild solution to Einstein's equations who pioneered this prediction/discovery, being the first pioneering solution to Einstein's field equations.) To learn and read more about the mathematics of General Relativity, and how Differential Geometry is extensively used in the theory and other areas of physics, here's some links: http://en.wikipedia.org/wiki/Mathematics_of_general_relativity (Granting the wikipedia article shows and discusses high level mathematics, so it might be hard to really get a grasp on them. I myself tried to read it, but lacking knowledge of tensors and differential geometry in general, makes it very difficult to understand) http://en.wikipedia.org/wiki/Differential_geometry#Applications Here's also the Wikipedia page of Differential Geometry To quote the Wiki, under the Applications section: 


#6
Jan1713, 04:29 AM

P: 43

By land survey, you mean: http://en.wikipedia.org/wiki/Surveying ? I must admit, it's the first time I heard about such a thing, atleast as a scientific field kind of sense. What is it about? 


#7
Jan1713, 08:01 AM

Sci Advisor
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#8
Jan1713, 08:07 AM

C. Spirit
Sci Advisor
Thanks
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#9
Jan1713, 11:33 PM

P: 530

http://www.youtube.com/watch?v=pUHu6...93237E&index=8 


#10
Jan1913, 08:10 AM

P: 43

0Hate to be technical but they aren't necessarily the shortest paths. They extremize but don't necessarily minimize.[/QUOTE]
Oh thank you for correcting me, and clearing it out! To quote the Wiki page of Differential Geometry, under the Applications section, the third item under Physics: 


#11
Jan1913, 10:21 AM

P: 308

To answer the first question: It's pretty unimportant compared to e.g. differentiation and integration in R^n, linear algebra or differential equations.



#12
Jan2013, 02:47 PM

P: 188

Differential geometry is also needed if one wants to understand the geometric setting of gauge theories which are formulated using a mathematical object called fiber bundles. In short, it is good to study differential geometry if you want to be a theoretical physicist.



#13
Jan2213, 04:21 AM

P: 43

Riemann, in turn, pioneered and develop his own branch of differential geometry, Riemannian Geometry, which is the key specific mathematics from Differential Geometry used in GR. To quote http://en.wikipedia.org/wiki/Riemannian_geometry: 


#14
Jan2513, 01:33 AM

P: 714

There is no way this is believable without some kind of reference. 


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