Page 202 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 17: Tensor Fields in Curvilinear Coordinates

Exercise 2.1: Calculate rot A, div H, grad  (vectorial gradient) in cylindrical and spherical Notes
coordinates.
Exercise 3.1: Calculate the Laplace operator  applied to the scalar field  in cylindrical and in
spherical coordinates.

17.10 Summary

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
3
2
1
whole space or at least in some domain, where we are going to use our coordinates y , y ,
2
1
y . In Cartesian coordinates this map P  (y , y , y ) is constructed by means of vectors
1
3
3
2
and bases. Arranging 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
1
2
and y =  are two angles. By the way, spherical coordinates are one of the simplest
3
examples of curvilinear coordinates. Furthermore, let’s keep in mind spherical coordinates
when thinking about more general and hence more abstract curvilinear coordinate systems.
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,
2
1
3
the best idea is to represent each point P by its radius vector r in some auxiliary Cartesian
P
coordinate system and then consider a map r  (y , y , y ). The radius-vector itself is
2
P
3
1
represented by three coordinates in the basis e , e , e of the auxiliary Cartesian coordinate
1
2
3
system:
3
P 
i
r  x e .
i
i 1

Therefore, we have a one-to-one map (x , x , x )  (y , y , y ). Hurrah! This is a numeric
3
3
2
1
1
2
map. We can treat it numerically. In the left direction it is represented by three functions
of three variables:
2
1
1
3
1
 x  x (y ,y ,y ),
 2 2 1 2 3
 x  x (y ,y ,y ),
 3 x (y ,y ,y ).
3
3
1
2
 x 
Cartesian basis vectors e do not depend on y ; therefore, they are not differentiated. Both
j
 q
sides are expansions in the base e , e , e of the auxiliary Cartesian coordinate system.
2
1
3
17.11 Keywords
Spherical coordinates are one of the simplest examples of curvilinear coordinates.
Vector-function: Then the vector-function
3
R(y , y , y ) = r = x (y ,y ,y )e i
3
i
P 
1
2
3
2
1
i 1

is a differentiable function of three variables y , y , y .
3
2
1
Christoffel symbols  form a three-dimensional array with one upper index and two lower
k
ij
indices.
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