Page 203 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes 17.12 Self Assessment

1. ................... are one of the simplest examples of curvilinear coordinates.

2. The ................... itself is represented by three coordinates in the basis e , e , e of the auxiliary
2
1
3
3
i
P 
Cartesian coordinate system r  x e .
i
i 1

3. Coordinate lines taken in whole form a coordinate grid. This is an infinitely dense grid.
But usually, when drawing, it is represented as a grid with ...................
4. The parallels do not intersect, but the ................... one family of coordinate lines do intersect
at the North and at South Poles. This means that North and South Poles are singular points
for spherical coordinates.
5. ...................  form a three-dimensional array with one upper index and two lower indices.
k
ij
6. ................... e do not depend on y ; therefore, they are not differentiated when we substitute
j
q
3  E 3 3  S q 3 3
k Both sides of
q 
j
q
i 

E  S e into i j   k ij E .  i j e   k ij S e are expansions in the
k q
j
i
i 1  y k 1 q 1 y k 1 q 1





base e , e , e of the auxiliary Cartesian coordinate system.
1 2 3
17.13 Review Questions
1. Remember the exact definition of spherical coordinates and find all singular points for
them.
3
2. Relying upon formula  E i j   k E , calculate the Christoffel symbols for cylindrical

 y k 1 ij k

coordinates.
3
3. Remember formula E   R i   x j e  3 S e from which you derive
j
j 

i
j
i
 y j 1 y i j 1


 R
E =  y i .
i
Answers: Self Assessment
1. Spherical coordinates 2. radius-vector
3. finite density. 4. meridians forming
5. Christoffel symbols 6. Cartesian basis vectors

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