Page 167 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes 15.1 Covectors

In previous sections, we learned the following important fact about vectors: a vector is a physical
object in each basis of our three-dimensional Euclidean space E represented by three numbers
such that these numbers obey certain transformation rules when we change the basis.

Now suppose that we have some other physical object that is represented by three numbers in
each basis, and suppose that these numbers obey some certain transformation rules when we
change the basis. Is it possible? One can try to find such an object in nature. However, in
mathematics we have another option. We can construct such an object mentally, then study its
properties, and finally look if it is represented somehow in nature.
Let’s denote our hypothetical object by a, and denote by a , a , a that three numbers which
2
3
1
represent this object in the basis e , e , e . By analogy with vectors we shall call them coordinates.
2
3
1
But in contrast to vectors, we intentionally used lower indices when denoting them by a , a , a .
3
1
2
Let’s prescribe the following transformation rules to a , a , a when we change e , e , e to
1
1
2
3
3
2
e ,e ,e :
  
1
3
2
3
i
j 

a  S a , ...(1)
j i
i 1

3
i
j 
a  T a , ...(2)
i
j
i 1

Note that (1) is sufficient, formula (2) is derived from (1).
Example: Using the concept of the inverse matrix T = S derive formula (2) from
–1
formula (1).
Definition: A geometric object a in each basis represented by a triple of coordinates a , a , a and
3
2
1
such that its coordinates obey transformation rules (1) and (2) under a change of basis is called a
covector.
Looking at the above considerations one can think that we arbitrarily chose the transformation
formula (1). However, this is not so. The choice of the transformation formula should be
self-consistent in the following sense. Let e , e , e and e ,e ,e  be two bases and let e ,e ,e  be
  
 
 
1
2
3
1
1
2
3
2
3
the third basis in the space. Let’s call them basis one, basis two and basis three for short. We can
pass from basis one to basis three directly. Or we can use basis two as an intermediate basis.
In both cases, the ultimate result for the coordinates of a covector in basis three should be the
same: this is the self-consistence requirement. It means that coordinates of a geometric object
should depend on the basis, but not on the way that they were calculated.
Exercise 15.1: Replace S by T in (1) and T by S in (2). Show that the resulting formulas are not
self-consistent.
What about the physical reality of covectors? Later on we shall see that covectors do exist in
nature. They are the nearest relatives of vectors. And moreover, we shall see that some
well-known physical objects we thought to be vectors are of covectorial nature rather than
vectorial.
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