Introduction to Vectors and Tensors Vol 2 (Bowen 246). - Index of

320 Chap. 9 • EUCLIDEAN MANIFOLDS
44.3 Spherical coordinates
1 2 3
( x , x , x ) are defined by the coordinate transformation
z = x sin x cos x
1 1 2 3
z = x sin x sin x
2 1 2 3
z = x cos x
3 1 2
1 2 3
relative to a rectangular Cartesian coordinate system ẑ . How must the quantity ( x , x , x )
1
be restricted so as to make ẑ xˆ
− one-to-one? Discuss the coordinate curves and the
coordinate surfaces. Show that
⎡1 0 0 ⎤
1 2
⎡g
( )
⎢
ij
0 ( x ) 0
⎥
⎣ x ⎤
⎦ =
⎢ ⎥
1 2 2
⎢⎣
0 0 ( x sin x ) ⎥⎦
44.4 Paraboloidal coordinates
1 2 3
( x , x , x ) are defined by
z = x x cos x
1 1 2 3
z = x x sin x
2 1 2 3
1
z = ( x ) − ( x )
2
( )
3 1 2 2 2
1 2 3
Relative to a rectangular Cartesian coordinate system ẑ . How must the quantity ( x , x , x )
1
be restricted so as to make ẑ xˆ
− one-to-one. Discuss the coordinate curves and the
coordinate surfaces. Show that
1 2 2 2
⎡( x ) + ( x ) 0 0 ⎤
⎢
1 2 2 2 ⎥
⎡
⎣gij( x ) ⎤
⎦ =
⎢
0 ( x ) + ( x ) 0
⎥
1 2 2
⎢ 0 0 ( xx)
⎥
⎣
⎦
44.5 A bispherical coordinate system
coordinate system by
1 2 3
( x , x , x ) is defined relative to a rectangular Cartesian