tensors for dummies

[Johanluus]

Can anybody help , with a basic tutorial for the

introduction to tensors - with a workable example , perhaps.

 

My algebra and calculus , is ok but slightly rusty.

[timo]

I think unless you have specific questions you're better off googling for an intro text, say for "introduction tensors", read it and ask here if you have specific questions. Someone here might be willing to give you an introduction here without you having asked a specific question. But it will certainly not reach the quality of stuff you find using Google.

[ajb]

Carroll's lecture notes on general relativity may be of help, depending on what level of sophistication you require.

[Bignose]

just tensors or all vector calculus? if the later, I cannot recommend Shey's Div, Grad, Curl and all that more.

[Johanluus]

Ive started on "Carroll's lecture notes on general relativity" so far so good , thanks all.

[ydoaPs]

just tensors or all vector calculus? if the later, I cannot recommend Shey's Div, Grad, Curl and all that more.

 

I don't remember tensors being a part of that book. Good book, though, from what I remember.

[Bignose]

I don't remember tensors being a part of that book. Good book, though, from what I remember.

 

Yeah, that's why I asked, because if someone needs vector calculus and tensors, then they should start with the vector calculus. Because a lot of the tensor stuff is just an extension of the vector stuff (not all, but a lot of it).

[Royston]

This is the best overview of what tensors actually 'are' that I've found...

 

http://www.grc.nasa.gov/WWW/k-12/Numbers/Math/documents/Tensors_TM2002211716.pdf

 

Carroll's lecture notes are hardly an introduction to tensors. GR is one application of tensors, there's many more basic examples that require tensors.

Edited June 25, 2010 by Snail

[ajb]

This is the best overview of what tensors actually 'are'

 

what tensors are can depend on your view point. To me they are examples of geometric objetcs on a manifold. These consist of

 

1. with respect to any allowable coordinate system there is one and only one ordered system of functions called components with respect to the given coordinate system.

 

2. a law which allows the representation of the components in an allowable coordinate system in terms of the components in any other allowable coordinate system, the corresponding coordinate transformations, the Jacobian matrix and their derivatives.

 

This I think is the most general definition of geometric objects, but clearly not the most elegant. Characteristics of a geometric object include the number of components, the highest order of derivatives in the transformation law and the particular representation of the groupoid of coordinate transformations the object forms. For example, tensor fields are first order objects and form a linear representation. Connections for example are second order and form an affine representation.

 

I only quote Carroll as he quickly spells out the transformation rules of tensors. It will depend on what one has in mind as applications as to how useful his presentation really is.

 

As an aside there is also the notion of a geometric object as a section of a natural bundle over the manifold. A natural bundle is a functor from the category of manifolds to the category of vector bundles such that local diffeomorphisms become vector bundle automorphisms. I question exactly how useful this is in supergeometry as we are faced with the realisation that not all "sensible " representations of the diiffeomorphisms are built in this way. However, the notion of a natural bundle is still very useful and extends to manifolds with more general gradings. Many be this is another story...

Edited June 26, 2010 by ajb