Maths - Covectors

On the vector page we saw that vectors take elements with a field structure and create an array of these field elements which produces a vector space structure. This is described as a vector 'over' a field.

  name operations operands
combined structure vector space addition and scalar multiplication scalar and vector
element structure field add,subtract, mult and divide number such as integer, real and so on.

A covector is the dual of this:

that is: if we think of a vector as a mapping from a field to a vector space then a covector represents a mapping from a vector space to a field.

So how can we construct a covector?

Linear Functional

A linear function like:

ƒ(x,y,z) = 3*x + 4*y + 2*z

has similarities to vectors, for instance we can add them for instance if:

ƒ1(x,y,z) = 3*x + 4*y + 2*z

and

ƒ2(x,y,z) = 6*x +5*y + 3*z

then we can add them by adding corresponding terms:

ƒ1(x,y,z) + ƒ2(x,y,z) = 9*x + 9*y + 5*z

We can also apply scalar multipication, for instance:

2*ƒ1(x,y,z) = 6*x + 8*y + 4*z

We can also multiply them together, for instance:

ƒ1(x,y,z) * ƒ2(x,y,z) = 18 x² + 39 xy + 20y² + 21 xz + 22 yz + 6z²

but this product is no longer linear but is quadratic.

So these functions have the same properties as vectors (that is they are isomorphic to vectors), however we now want to reverse this and make the functions ƒ1and ƒ2 the unknowns and the vector is the known:

  x   x   x   x  
(a*ƒ1+ b*ƒ2+ c*ƒ3)( y )=(a*ƒ1)( y )+(b*ƒ2)( y )+(c*ƒ3)( y )
  z   z   z   z  

So if we supply values for the vector what is the unknown function (or the function multipliers a,b and c) ?

3   7   8   4  
4 =( 3 )+(b*ƒ2)( 4 )+(c*ƒ3)( 4 )
2   1   2   6  

Duals

We can generalise this duality between vectors and covectors to tensors one of the aims of this type of approach is to analyze geometry and physics in a way that is independent of the coordinate system.

The duality shows itself in various ways :

  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 infinitesimally 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.

Coordinate Independence

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.

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:

e et= [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:

et e = 1

because:

e0
e1
e0 e1
=
e0•e0+e1•e1
=
1

but a scalar multiplication by 1 is the same as a matrix multiplication by [I] so we have: et e = e et= [I] = 1 = δij

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

aT a = [I]

a0 a1
b0
b1
=
a0*b0 + a1*b1

where:

We can combine these terminologies to give:

(aT a)ij = eki • ekj = δij

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?

 


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