differential geometry lecture notes

**franznietzsche** · Dec2-03, 01:11 AM

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.

**franznietzsche** · Dec2-03, 08:37 PM

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?

**lethe** · Dec2-03, 08:46 PM

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.

**Ambitwistor** · Dec2-03, 08:49 PM

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.

**Loren Booda** · Dec2-03, 08:52 PM

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.

**franznietzsche** · Dec2-03, 09:57 PM

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.

**rdt2** · Jan21-04, 04:17 PM

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.

Jan24-04, 04:58 AM

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_wor...tro_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 e_k. If the components of a vector A are A^k then

A = A^k e_k

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]

**Steven S** · Jan29-04, 10:43 AM

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.

**Ebolamonk3y** · Apr26-04, 12:25 AM

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..

**franznietzsche** · Apr26-04, 09:46 AM

Originally Posted by Ebolamonk3y

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.

**Ebolamonk3y** · Apr26-04, 05:47 PM

:) alright. Thanks.

**Ebolamonk3y** · Apr26-04, 05:49 PM

btw... which ones did you get from Amazon.com?

**Stevo** · Apr27-04, 02:10 AM

I found Tensor Geometry by Dodson & Poston to be a good book.

**franznietzsche** · Apr27-04, 02:27 AM

Originally Posted by Ebolamonk3y

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...

**H-bar None** · Apr28-04, 06:19 PM

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.