Differential geometry and tensor calculus

[hawaiidude]

i have been doing fourier, differential equations, and advanced calculus and then i saw differential geometry in a book...since the book only covered advanced calculus, it only introduced diff. geometry...can anyone show and tell me where on the web there is a tutorial for differential geometry?

 

[quartodeciman]

DiffGeo combines calculus and geometry. While it invades geometry with the methods of calculus, it also advances calculus onto richer geometric frameworks. For background, the equivalent understanding of a calculus and differential equations survey is desirable (I see you have that!). Also, since it gets into research level areas, a familiarity with the grammar and basic content of abstract algebra and general topology is desirable. Much of the online instructional material is physics-oriented. That covers much of the historical (nineteenth century to early twentieth century) interest of the subject, but not so much the later pure mathematical interest, found mostly in books and journals.

Here are a few links:

the position of DiffGeo within the mathematics world ->
http://www.math.niu.edu/~rusin/known-math/index/tour_geo.html
Math Atlas: geometric areas of mathematics

a short introduction to DiffGeo terms ->
http://www.wikipedia.org/wiki/Differential_geometry
Wikipedia: Differential Geometry

two physics-oriented online sets of lecture notes ->
http://people.hofstra.edu/faculty/Stefan_Waner/diff_geom/tc.html
Introduction to Differential Geometry and General Relativity

http://people.uncw.edu/lugo/COURSES/DiffGeom/dg1.htm
Differential Geometry and Physics

(don't forget to enjoy what you are doing)

 

[hawaiidude]

hey thanks man those links really helped me...our of curiosity, can differential geometry be apllied to worm holes and black holes?

 

[quartodeciman]

The U. Cambridge (S. Hawking and N. Turok territory!) Tripos cirriculum says this in their description:

"Differential Geometry applied to Physics, with applications to Maxwell Theory, General Relativity, Quantum Mechanics and parts of Quantum Field Theory, String Theory and M. Theory"

I also notice most of the applicable physics course descriptions do NOT specify DiffGeom as a necessary prerequisite, but promise to develop what is needed in the body of those courses. They probably want to emphasize that the physics preparation be adequate.

You might think more along lines of what physics tools-of-the-trade are needed. Mathematical techniques are apt to be an eclectic grab-bag.

link ->

http://www.maths.cam.ac.uk/CASM/courses/descriptions/
U. Camb. Certificate of Advanced Study in Mathematics

 

[rick1138]

Sean Carroll's Lecture Notes on General Relativity have quite a bit of DG - advanced, though very detailed and illuminating:

http://pancake.uchicago.edu/~carroll/notes/

 

[hawaiidude]

i can't open it...how do i open the files> real one player?

 

[quartodeciman]

Carroll's files are PostScript files. You might need something like Ghost Script and Ghost Script Viewer.

link -->
http://www.cs.wisc.edu/~ghost/index.html