Maths - Covectors

The idea of covectors comes from the concept of tensors and one of the things we are trying to to do is analyse geometry and physics in a way that is independent of the coordinate system.

When I search on the web for the idea of 'covectors' every site I find seems to have a different way to introduce them. There seem to be at least six approaches:

1. If vectors are related to columns of a matrix then covectors are related to the rows.
2. The dot product of a vector and its corresponding covector gives a scalar.
3. When the coordinate system in changed then the covectors move in the opposite way to vectors (Contravariant and Covariant).
4. If a vector is made from a linear combination of basis vectors then a covector is made by combining the normals to planes.
5. When we take an infinitseimally small part of a manifold the vectors form the tangent space and the covectors form the cotangent space.
6. vector elements are represented by superscripts and covector elements are represented by subscripts.

I would like to try to reconcile these approaches and work out which is most fundamental.

It seems to me that one of the most fundamental things about tensors is about changes to the coordinate system and defining things in a coordinate independant way and therefore I would like to approach this from that point of view.

So lets start with a 3D global orthogonal 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_ieⁱ in the red coordinate system and

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

where:

• p = physical vector being represented in tensor terms
• v_i = tensor in the red coordinate system
• eⁱ= 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:

e e^t= [I]

For example, in the simple two dimensional case:

e0 e1

e0 e1

=

e0•e0 e0•e1 e1•e0 e1•e1

=

1 0 0 1

since the basis vectors are orthogonal then:

e₀•e₀ = e₁•e₁ = 1
e₁•e₀ = e₀•e₁ = 0

also:

e^t e = 1

because:

e0 e1

e0 e1

=

e0•e0+e1•e1

=

1

but a scalar multipication by 1 is the same as a matrix multipication by [I] so we have: e^t e = e e^t= [I] = 1 = δ_ij

Now instead of looking at the basis vectors we will look at

a^T a = [I]

a0 a1

b0 b1

=

a0*b0 + a1*b1

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.

Non-Orthogonal Coordinates

So far we assumed the coordinates are linear and orthogonal but what if the coordinates are curviliner?

 

Directed Area

Duality