# Quick Introduction to Tensor Analysis

1. Mar 17, 2004

Staff Emeritus
Ruslan Sharipov has a nifty online textbook on this subject. It's written in interactive do-it-yourself style. Give it a glance, and see what you think.

2. Mar 17, 2004

### pmb_phy

3. Mar 22, 2004

### 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

4. Mar 24, 2004

### 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.

Last edited: Mar 24, 2004