Page 195 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes However, by substituting (3) into the arguments of S , or by substituting (2) into the arguments
i
j
of T , we can make them have a common set of arguments:
i
j
1
i
3
2
i
3
i
i
2
S , S (x ,x ,x ), T  T (x ,x ,x ), ...(6)
1

j
j
j
j
1
2
2
3
i
i
S , S (y ,y ,y ), T  T (y ,y ,y ), ...(7)
3
i
i
1

j
j
j
j
When brought to the form (6), or when brought to the form (7) (but not in form of (5)), matrices
S and T are inverse of each other:
T = S . ...(8)
–1
This relationship (8) is due to the fact that numeric maps (2) and (3) are inverse of each other.
Exercise 1.1: You certainly know the following formula:
3
1
df(x (y), x (y), x (y)) 3 f (x (y),x (y),x (y)) dx (y) ' f  .
i
2
3
1
2
'
i
dy   i dy , where f   x i
i 1

It’s for the differentiation of composite function. Apply this formula to functions (2) and derive
the relationship (8).
17.3 Coordinate Lines and the Coordinate Grid
Let’s substitute (2) into (1) and take into account that (2) now assumed to contain differentiable
functions. Then the vector-function
3
R(y , y , y ) = r = x (y ,y ,y )e i ...(1)
i
3
2
1
P 
3
1
2
i 1

is a differentiable function of three variables y , y , y . The vector-function R(y , y , y ) determined
2
3
1
2
3
1
by (1) is called a basic vector-function of a curvilinear coordinate system. Let P be some fixed
0
point of space given by its curvilinear coordinates y ,y ,y . Here zero is not the tensorial
1
2
3
0
0
0
index, we use it in order to emphasize that P is fixed point, and that its coordinates y ,y ,y are
3
2
1
0
0
0
0
three fixed numbers. In the next step let’s undo one of them, say first one, by setting
2
3
1
2
1
3
y = y  t, y = y , y = y . ...(2)
0
0
0
Substituting (2) into (1) we get a vector-function of one variable t:
3
1
2

R (t) R(y  t,y ,y ). ...(3)
0
0
1
0
If we treat t as time variable (though it may have a unit other than time), then (3) describes a
curve (the trajectory of a particle). At time instant t = 0 this curve passes through the fixed point
P . Same is true for curves given by two other vector-functions similar to (4):
0
1
2
3
R (t) = R(y ,y  t,y ), ...(4)
2 0 0 0
2
1
3
R (t) = R(y ,y ,y  t). ...(5)
3
0
0
0
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