Differential geometry Definition and 424 Threads

Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century.
Since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds. Differential geometry is closely related to differential topology and the geometric aspects of the theory of differential equations. The differential geometry of surfaces captures many of the key ideas and techniques endemic to this field.

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  1. N

    Discover the Beauty of Differential Geometry: A Visual Guide with Notes

    Does anyone know where I can find a nice set of notes accompanied with pictures on Differential Geometry?
  2. I

    Should i take differential geometry?

    Should I take a semester-long differential geometry course (in lieu of a thermo/stat mech course, which while not terribly challenging would prepare me for qualifiers, etc), or should I just jump into GR and maybe pick up a mathematical methods in physics course later? I first learned partial...
  3. M

    Proving an Identity from Differential Geometry

    One often encounters the following identitiy in Tensor Analysis/Differential Geometry: dx^j (partial/partial x^i) = partial x^j / partial x^i = delta ij It's easy to see why partial x^j / partial x^i = delta ij but how does dx^j (partial/partial x^i) = partial x^j/partial x^i ? I...
  4. R

    Defining Euclidean Norm in Phase Space: A Differential Geometry Analysis

    Hi to everynoe! I have a bit of trouble in understanding the following thing : Suppose we have a phase space, in which a dynamical system evolves: for example a two dimensional vector space: temperature and time. Now, does it make a sense to define the euclidean norm of a vector in such...
  5. F

    Problem I'm having with intrinsic equations when studing differential geometry

    I've just started learning about intrinsic equations after learning about envelopes, arc length and curved surface area in cartesian, parametric and polar coordinates, and understand the principle behind intrinsic equations, but my book doesn't explain it particularly well, here is a question...
  6. JasonRox

    Introduction to Differential Geometry

    So, where do I begin? I have a current interests in going into Algebraic Topology and I am currently learning about Point-Set Topology and Abstract Algebra, which are pre-requisites to Algebraic Topology. After talking to a professor, he recommended learning some Differential Geometry, but...
  7. R

    Worked Problems in Modern Differential Geometry

    I'm looking for resources of worked problems in modern differential geometry. There are plenty that have theorems and general formulas but very few that gon into the dirty details of how to solve actual problems. Any information would be appreciated.
  8. A

    Advice for Taking Differential Geometry in College

    I am a senior in HS right now. I am taking my classes at a local university because I finished my HS classes early, but will still graduate with my class this June. I am taking Differential Equations now, but would like to take a geometry class next semester. The prereqs are multivariable...
  9. L

    Why is differential geometry not required in undergraduate physics curriculum?

    differential geometry... I am constantly being told that this is important stuff for undergradautes to know about and it is taught by the end of the secon year...however, I am not going to be seeing ANY of this stuff at all during undergraduate school. And there isn't even a class offered in...
  10. A

    What's in your differential geometry toolbox?

    I saw GRTensorII mentioned here awhile back. What are the most powerful tools out there (which still run on a PC), including user manuals and texts, that you would recommend for working with field equations, curvature tensors, line elements, etc?
  11. E

    How to Expand a Vectorial Function into a Taylor Series?

    I'm trying to demonstrate the following proposition: Let \vec{\alpha}(s) be a natural parametrization of an arc C. Then: \vec{\alpha}(s+h)=\vec{\alpha}(s)+\left(h-\frac{\kappa^2h^3}{6}\right)\hat{t}+\frac{1}{2}\left(\kappa...
  12. kakarukeys

    Please recommend differential geometry books

    I want to learn the mathematical language of Ashtekar's formulation / Loop Quantum Gravity in 3 months. Which introductory differential geometry texts do you recommend? I prefer books that are more mathematical, such as those that start from definitions, theorems, proofs, etc but not too...
  13. Q

    Tensors & Differential Geometry - What are lie groups?

    Tensors & Differential Geometry -- What are lie groups? I've heard a lot about "lie groups" on this section of the forum, and was wondering what they are and if someone could explain it in simple terms. Thank you.
  14. H

    Prerequisites for Differential Geometry

    Hello, I was wondering what you guys think is the absolute minimum requirements for learning Differential Geometry properly and also how would you go about learning it once you got to that point, recommended books, websites, etc. I am learning on my own because of some short circuit in my brain...
  15. M

    Best books for Differential Geometry?

    What are the best books for learning differential geometry well? Any recommendations appreciated.
  16. A

    Differential Geometry And Difference Geometry?

    HELLO ALL, I AM StUDYING DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS IN IMAGE PROCESSING. WHAT EVERYBODY DOES IS THAT TREATS THE IMAGE OR OBJECT AS A CURVE OR A SURFACE AND USES VARIOUS DIFFERENTIAL GEOMETRY OPERATORS TO GET THE DESIRED RESULT, FIRST IN THE CONTINUOS DOMAIN AND THEN FOR...
  17. L

    Most efficient backgrounder on differential geometry ?

    I am used to classical tensor notations. I am doing theoretical physics during my hobby time, engineering for a living. Often I get lost with differential forms notations and even I don't recognize easily the concepts. I don't have the patience to re-read the full text of the wonderful...
  18. S

    What Starter Texts Can Help Understand Differential Geometry for GTR?

    Hello, I was hoping if somebody could point me to a starters text on differential geometry. I have done some calculas ~4 years ago - I can refresh stuff as I need to. My physics is at high school level, but I do read up on stuff every once a while My aim in learning diff geom is to be...
  19. F

    Differential geometry lecture notes

    i've been trying to find a good set of lecture notes for independent study on the subject. I wen to the one in the thread on differential geometry and tensor calculus at people.hofstra.com, but it went offline while i was viewing it and i have had no luck reaccessing it.
  20. marcus

    Proposed Resource Sticky for Quantum Differential Geometry

    Quantum Geometry is a fairly new field using a wide range of math tools. It would help to have a sticky where links to useful online papers can be kept for ready access. A lot of the interest in quantum differential geometry stems from efforts to quantize the 3+1 spacetime manifold. Not to...
  21. marcus

    Basic tools for quantized differential geometry

    One of selfAdjoint's posts in a loop gravity thread pointed out that many of the tools needed belonged in differential geometry. This is his 22 October 8:33 pm post: ----------------------- "Since Greg the magician got us a reprieve on the bandwidth thing, I'll add a little. I just read...
  22. H

    Calculating Wormholes: Scientists and Differential Geometry

    HOW DO SCIENtists calculate worm holes? differential geometry? what do they do? i know a worm has 4d xyz and time...oike what do they do? the distance from Earth to a worm and then do mental math?
  23. H

    Differential geometry and tensor calculus

    i have been doing fourier, differential equations, and advanced calculus and then i saw differential geometry in a book...since teh 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?
  24. D

    How Are Surface Curvatures Derived from Implicit and Parametric Forms?

    I am trying to learn a little bit more about geometry of surfaces and some differential geometry concepts like principal, gaussian an mean curvatures. I have found some interesting material at mathworld.wolfram.com but as usual there is a quantity of little incorrect details. Is anybody...
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