Differential geometry lecture notes

In summary: Tensor Analysis on Manifolds by Jayant V. NarlikarTensor Geometry by Dodson & PostonI found Tensor Geometry by Dodson & Poston to be a good book.
  • #1
franznietzsche
1,504
6
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.
 
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  • #2
it went back up online again, so i don't need help with that, but can someone please explain the use of the superscript notation for the local coordinates of the manifolds? It looks like it means x to the first power, x to the second power, etc. but that doesn't seem to make sense (because it is never referred to as a power). So is it a power function or is it simply like a subscript, excpet a superscript instead?
 
  • #3
Originally posted by franznietzsche
So is it a power function or is it simply like a subscript, excpet a superscript instead?

It is not an exponent. it is just a label, like a subscript, but its a superscript instead. the reason for this is to allow the Einstein summation notation, which is quite nice, but may take some getting used to.
 
  • #4
It's not an exponent (power), it's just an index, like a subscript index. I haven't seen this nomenclature before, but they appear to be using subscripts to denote extrinsic coordinates in the embedding space, and superscripts to denote intrinsic coordinates in the surface.
 
  • #5
A simple introduction to the Einstein notation and its basic applications to general relativity can be found in Introduction to Cosmology by Jayant V. Narlikar.
 
  • #6
ok thnax that helps. This is my first experience with anything beyond Fourier series and differential equations so that usage of the superscript was entirely foreign to me. Thanx for the help.
 
  • #7
superscripts

Since superscripts are used to denote components of vectors, a notation I've seen (and since adopted) is to show powers in parentheses, thus:

x2 is the component of a vector x in the 2-direction

x(2) is x*x.

Cheers,

Ron.
 
  • #8
Originally posted by franznietzsche
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.


See
http://www.math.odu.edu/~jhh/counter2.html
http://xxx.lanl.gov/abs/gr-qc/9712019
http://arcturus.mit.edu/8.962/notes.html
http://www.geocities.com/physics_world/ma/intro_tensor.htm

The superscripts are placed as they are in order to use the summation convention and to distinguish between two different types of objects.

A unit vector is written as ek. If the components of a vector A are Ak then

A = Ak ek

where the summation convention is used:
[qoute]
Summation Convention: If an index appears twice in a term, once as a superscript and once as a subscript, then summation is implied over the range that the indices are allowed to take on.
[/quote]
 
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  • #9
I recommend Bishop and Goldber'g "Tensor Analysis on Manifolds". It is really cheap since it is a Dover book and covers all the basics.
 
  • #10
What are the pre-reqs and such for Tensor analysis?

What other materials do you guys recommend for introduction to tensor analysis?

BTW, is there such a volume purchasing for Dover books or Schaums outlines? Like, can you purchase a ton of them for a discount price? Thanks in advance..
 
  • #11
Ebolamonk3y said:
What are the pre-reqs and such for Tensor analysis?

What other materials do you guys recommend for introduction to tensor analysis?

BTW, is there such a volume purchasing for Dover books or Schaums outlines? Like, can you purchase a ton of them for a discount price? Thanks in advance..

I recently purchased several from Amazon.com which had the best prices i could find (several dollars less than any other retailer).

I haven't started Tensor Analysis on manifolds (the Goldberg one), but Lovelock and Rund's Tensors, Differential Forms, and variational Principles is a very good book. The two in combination give a very nice treatment of the subject.

As for prereqs, reading the books you just need to know calculus, partial differentiation (which is just a very easy extension of one variable differentiation) and an understanding of vector analysis will help, but is not necesary as that is covered in the text.
 
  • #12
:) alright. Thanks.
 
  • #13
btw... which ones did you get from Amazon.com?
 
  • #14
I found Tensor Geometry by Dodson & Poston to be a good book.
 
  • #15
Ebolamonk3y said:
btw... which ones did you get from Amazon.com?


Well pretty much everything is cheaper there, at least if you're looking for physics/mathematics texts. The ones i got were Tensors, Differential Forms and Variational Principles by Lovelock and Rund and i also got Tensor Analysis on manifolds the authors of which i don't feel like looking up midnight, though i think itd Goldberg and someone else...
 
  • #16
Tensors, Differential Forms and Variational Principles by Lovelock and Rund is an excellent book. I've spent a lot of time reading books on tensors lately the light bulb lit up on this one.

Shaum's has a book on Vector Analysis and Tensors that is pretty good.
 
  • #17
Thanks for the posts and references. I want to issue a warning though on some of these. the free notes available at

http://www.math.odu.edu/~jhh/counter2.html

for example, seem devoted primarily to the old fashioned, index - dominated non conceptual approach, ignoring the meanings of the objects being discussed, at least in the first chapter I read, introducing tensors. maybe it gets more conceptual later.

This "indices only" approach however has been out of date for over 40 years in mathematical circles, and it was my impression also, that even in the book on relativity by the physicists J.A. Wheeler and Kip Thorne, written in the 70's, the modern viewpoint was used.


The site at wikipedia suggested by someone in the thread What is a tensor?, has both approaches, and tries to bridge the gap, while pointing out that the conceptual approach is pretty much the standard in advanced work nowadays.

Bear in mind that I am a mathematician, and not a physicist, but you might check out which language the modern physicists are using more today and try to learn that.

One remark is that the conceptual does allow you to understand the indexed approach eventually, since it is just the coordinate representation of the conceptual version, but the index only approach seems to make it very hard to understand the conceptual approach, because no abstract concepts have been introduced.

I.e. the conceptual approach does contain the other, but not vice versa.
 
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  • #18
PS: The second reference in post 8 by Arcon, to notes of Sean Carroll of the Institute for Theoretical Physics at Santa Barbara, does explain both approaches.

http://xxx.lanl.gov/abs/gr-qc/9712019
 
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  • #19
Lecture notes about differential geometry

Hi
you can find different lecture notes about differential geometry
here:

http://freescience.info/books.php?id=38

Cheers
Cla
 

1. What is differential geometry?

Differential geometry is a branch of mathematics that studies the properties of curves and surfaces in higher-dimensional spaces. It uses techniques from calculus, linear algebra, and topology to understand the geometric structure of these objects.

2. What are some applications of differential geometry?

Differential geometry has many practical applications in physics, engineering, and computer science. It is used to model the shape of objects in computer graphics, to analyze the motion of particles in physics, and to study the curvature of spacetime in general relativity.

3. What are the main concepts in differential geometry?

The main concepts in differential geometry include curves, surfaces, manifolds, tangent spaces, and curvature. These concepts are used to describe and analyze the geometric properties of objects in higher-dimensional spaces.

4. How is differential geometry related to other branches of mathematics?

Differential geometry has connections to many other branches of mathematics, including topology, differential equations, and algebraic geometry. It also has applications in fields such as mechanics, thermodynamics, and optimization.

5. What is the difference between differential geometry and classical geometry?

Classical geometry focuses on the study of shapes and figures in the plane and in three-dimensional space, using Euclidean geometry as a basis. Differential geometry, on the other hand, extends this concept to higher-dimensional spaces and introduces the use of calculus and other mathematical tools to analyze the geometry of these spaces.

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