Maths - Changing Coordinates

An important topic for tensors is the way that they are affected by a change in coordinates. Also can we formulate our tensor equations in such a way that they are independent of the chosen coordinate system?

First we will start with a coordinate system based on a linear combination of orthogonal basis vectors.

change coordinates

The physical vector 'p' can be represented by either:

p = ∑ viei in the red coordinate system and

p = ∑ v'ie'i in the green coordinate system.

where:

So we can transform between the two using:

∑ v'k = tki v i

or

∑ ek = t'ki e'i

where:

Orthogonal Coordinates

We are considering the situation where a vector is measured as a linear combination of a number of basis vectors. We now add an additional condition that the basis vectors are mutually at 90° to each other. In this case we have:

ei • ej = δij

where:

If we choose a different set of basis vectors, but still perpendicular to each other, say e'i and e'j then we have:

e'i • e'j = δij

To add more dimensions we can use:

eki • ekj = δij

This is derived from the above expression using the substitution property of the Kronecker Delta.

We could express the above in matrix notation:

aT a = [I]

where:

We can combine these terminologies to give:

(aT a)ij = eki • ekj = δij

Any of these equations defines an orthogonal transformation.

Curvilinear Coordinate Systems

cylindrical coordinates

The simplest case contravarient and covarient tensors is to look at contravarient and covarient vectors. These

For a discussion about measuring the curvature of space, see this page.

Dyads

The tensor product of two vectors ei and ej may be denoted by ei ej or the operation may be shown explicitly by ei tensor product ej

A tensor can then be represented as:

∑ aij ei ej

 


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Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

cover us uk de jp fr ca Schaum's Outline of Theory and Problems of Tensor Calculus - I'm finding this hard going, it starts off with as review of linear algebra, matrix notation, etc. It redefines a lot of conventions which are hard to relearn, such as superscrips instead of subscripts to identify elements, and a summation convention, then it goes into coordinate transformations. I cant find any definition of what a tensor is. I think this book is aimed at people who already have some knowledge of the subject.

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