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.

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

p = ∑ v_{i}e^{i} in the red coordinate system and

p = ∑ v'_{i}e'^{i} in the green coordinate system.

where:

- p = physical vector being represented in tensor terms
- v
_{i}= tensor in the red coordinate system - e
^{i}= basis in the red coordinate system - v'
_{i}= tensor in the green coordinate system - e'
^{i}= basis in the green coordinate system

So we can transform between the two using:

∑ v'^{k} = t^{k}_{i} v ^{i}

or

∑ e^{k} = t^{'k}_{i} e'^{i}

where:

- t = a matrix tensor which rotates the vector v to vector '
- t' = a matrix tensor which rotates the basis e to the basis e'

## 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:

e_{i} • e_{j} = δ_{ij}

where:

- e
_{i}= a unit length basis vector - e
_{j}= another unit length basis vector perpendicular to the first. - δ
_{ij}= Kronecker Delta as described here.

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} = δ_{ij}

To add more dimensions we can use:

e_{ki} • e_{kj} = δ_{ij}

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

We could express the above in matrix notation:

a^{T} a = [I]

where:

- a = a vector
- a
^{T}= transpose of 'a' - [I] = the identity matrix

We can combine these terminologies to give:

(a^{T} a)_{ij} = e_{ki} • e_{kj} = δ_{ij}

Any of these equations defines an orthogonal transformation.

## Curvilinear Coordinate Systems

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

- The different ways that they vary with a change in coordinates shown by the partial differentials below.
- As duals of each other - for example in 3D vectors and bivectors are duals.
- Represented as column or row vectors.

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

## Dyads

The tensor product of two vectors e_{i} and e_{j} may be denoted by e_{i} e_{j} or the operation may be shown explicitly by e_{i} e_{j}

A tensor can then be represented as:

∑ a_{ij} e_{i} e_{j}