Page 194 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 17: Tensor Fields in Curvilinear Coordinates

17.1 General idea of Curvilinear Coordinates Notes

What are coordinates, if we forget for a moment about radius-vectors, bases and axes? What is
the pure idea of coordinates? The pure idea is in representing points of space by triples of
numbers. This means that we should have one to one map P  (y , y , y ) in the whole space or
3
2
1
at least in some domain, where we are going to use our coordinates y , y , y . In Cartesian
1
3
2
coordinates this map P  (y , y , y ) is constructed by means of vectors and bases. Arranging
1
2
3
other coordinate systems one can use other methods. For example, in spherical coordinates
y = r is a distance from the point P to the center of sphere, y = q and y =  are two angles. By the
1
2
3
way, spherical coordinates are one of the simplest examples of curvilinear coordinates.
Furthermore, let’s keep in mind spherical coordinates when thinking about more general and
hence more abstract curvilinear coordinate systems.
17.2 Auxiliary Cartesian Coordinate System
Now we know almost everything about Cartesian coordinates and almost nothing about the
abstract curvilinear coordinate system y , y , y that we are going to study. Therefore, the best
3
2
1
idea is to represent each point P by its radius vector r in some auxiliary Cartesian coordinate
P
system and then consider a map r  (y , y , y ). The radius-vector itself is represented by three
1
3
2
P
coordinates in the basis e , e , e of the auxiliary Cartesian coordinate system:
3
2
1
3
P 
i
r  x e . ...(1)
i
i 1

Therefore, we have a one-to-one map (x , x , x )  (y , y , y ). Hurrah! This is a numeric map.
2
3
2
1
1
3
We can treat it numerically. In the left direction it is represented by three functions of three
variables:
2
3
1
1
1
 x  x (y ,y ,y ),
 2 2 1 2 3
 x  x (y ,y ,y ), ...(2)
 3 x (y ,y ,y ).
3
3
2
1
 x 
In the right direction we again have three functions of three variables:
1
2
1
3
1
 y  y (x ,x ,x ),
 2 2 1 2 3
 y  y (x ,x ,x ), ...(3)
 y  y (x ,x ,x ).
3
3
2
1
3

Further we shall assume all functions in (2) and (3) to be differentiable and consider their partial
derivatives. Let’s denote
i
i
S   x i , T   y i . ...(4)
j
j
 y j  x j
Partial derivatives (4) can be arranged into two square matrices S and T respectively. In
mathematics such matrices are called Jacobi matrices. The components of matrix S in that form,
as they are defined in (4), are functions of y , y , y . The components of T are functions of
2
3
1
x , x , x :
2
1
3
3
3
2
i
2
1
i
1
i
S  S (y ,y ,y ), T (x ,x ,x ). ...(5)
j
j
j
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