Quick Introduction to Tensor Analysis

[selfAdjoint]

Ruslan Sharipov has a nifty online textbook (http://arxiv.org/abs/math.HO/0403252) on this subject. It's written in interactive do-it-yourself style. Give it a glance, and see what you think.

[pmb_phy]

Originally posted by selfAdjoint
Ruslan Sharipov has a nifty online textbook (http://arxiv.org/abs/math.HO/0403252) on this subject. It's written in interactive do-it-yourself style. Give it a glance, and see what you think.

I wrote this one up as an intro

http://www.geocities.com/physics_world/ma/intro_tensor.htm

[meteor]

Thanks, Selfadjoint, I guess that now I comprehend better what tensors are. I printed the document out.
I comprhend what vectors and covectors are, and comprhend the rules of transformations between different bases. ALso, more or less have an idea about what linear operators and bilinear forms are. I have problems comprhending the rules of transformations of linear operators between different bases, I refer explicitly to page 20, that says that a linear operator F_{j}^{i} transforms to another basis as


\bar{F}_{j}^{i} = \sum_{p=1}^{3} \sum_{q=1}^{3}
{T_{p}^{i} S_{j}^{q} F_{q}^{p}}
]


So, how do you get to the Ti p,Sq j and Fp q in the right side of the equality? I feel that i'm on the brim to completely understand tensor calculus, only have to work in a little details

[pmb_phy]

When learning tensor analysis/differential geometry it should be noted that there are two quite different things which are called "components" of a vector. The difference has to do with the difference between a natural (aka coordinate) basis and a non-natural basis. Unfortunately I haven't created a web page for this yet but its not that difficult to describe.

Consider the vector displacement dr in an N-dimensional Euclidean space. Using the chain rule this can be expanded to read

d\mathbf {r} = \frac {\partial \mathbf {r}} {\partial x^{i}} dx^{i} = dx^{i} \mathbf {e}_{i}

where

\mathbf {e}_{i} = \frac {\partial \mathbf {r}}{\partial x^{i}}

These form a set of vectors in which all other vectors may be expanded (i.e. a basis). These basis vectors are called natural basis vectors aka coordinate basis vectors. These basis vectors are not always unit vectors.