Contravariance and Covariance
Part 3 -- 9/28/2009
Vector Concept 3 -- Classical Tensor Analysis in Rn
Our transformations of coordinates have hitherto mostly involved constants
in the change of coordinates transformation matrix. We now extend the allowed transformations to include more
general functions, from which we
get yet another definition of a vector.
It will turn out, for general
non-cartesian coordinates, there are two possible transformation laws. One law describes
contravariant vectors, and the other covariant vectors.
Curvilinear Coordinates
Let x1,...,xn be a rectangular coordinate system in Rn.
(It should perhaps be remarked that it in general mathematical spaces, it is not
always possible to find such a coordinate sytem. The existence of a global rectangular
coordinate system is equivalent to assuming the space is Euclidean.)
Consider the general transformation of coordinates to a new, not necessarily rectangular,
coordinate system y1,...,yn:
y1 = y1(x1,...,xn)
... ...............
... ...............
... ...............
yn = yn(x1,...,xn)
|
(Eq. 1) |
We require certain technical conditions in order that our new coordinate system be
well defined. In
particular, the transformation should be differentiable
and have a differentiable inverse. Sufficient conditions for this are
given by the implicit function theorem -- the Jacobian of the transformation
should be non-zero. The y-coordinate
system so defined
constitutes a
curvilinear coordinate system.
Definition: A function f(x1,...,xn)
is a scalar if under a change of coordinates (1), it does not change value.
In other words, if f' is the value in the new y-coordinate system,
f'(y1,...,yn) =
f(x1,...,xn)
Typical examples of scalars are mass and temperature, which are independent of choice
of coordinates. A scalar is also called a tensor
of rank 0. Tensors of rank 1 are vectors, which we now define.
Definition: A contravariant vector is a set
of functions vk that, under a change of coordinates, transform
as
 |
(Eq. 2) |
where the Einstein summation convention has been used to indicate a summation over
the repeated index l.
Definition: A covariant vector is a set of
functions vk that, under a change of coordinates, transform as
 |
(Eq. 3) |
where the Einstein summation convention has been used to indicate a summation over
the repeated index l.
The origin of the above definitions may not at first be obvious. However, if we
expand the standard position "vector" P(x1,...xn) in the standard
rectangular coordinate basis {i1,...,in} ,the relation to our previous definitions becomes more apparent.
 |
(Eq. 4) |
we see the rectangular basis "vectors", {
i1,...,
in},
are given by
 |
(Eq. 5) |
Now, changing coordinates to y-coordinate system, we find
 |
(Eq. 6) |
Examining this equation for
ik, we recognize
 |
(Eq. 7) |
as the tangent vector to the y
m coordinate curve. The set of vectors
gm are known as the curvilinear basis vectors for the y-coordinate
system, and form a basis for R
n (That they form a basis
is gauranteed by virtue of the fact that the Jacobian of the transformation is non-zero). Therefore, any vector
v in R
n can be expanded as
 |
(Eq. 8) |
The v
k are the contravariant components of
v.
Denoting the dual basis to the {
gm} by {
gm},
we can expand
v in the dual basis as
 |
(Eq. 9) |
The v
k are the covariant components of
v.
It might come as a surprise that in the most general case in this definition, when
non-orthognal transformations are allowed, a finite directed line segment will not
in general obey the transformation laws (2) or (3) and so is NOT a vector. However, infinitesimal directed line segments
are vectors.
This whole analysis has assumed the existence of a global rectangular coordinate
system, or in other words, that the space we are considering is Euclidean. Easing
this restriction, and considering non-Euclidean spaces, leads us to the study of
vectors defined on more general spaces called manifolds.
Next week -- Vectors on Manifolds.