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

322 Chap. 9 • EUCLIDEAN MANIFOLDS
( − )
2 2 1 2 2
⎡a cosh x cos x
0 0 ⎤
⎢
⎥
2 2 1 2 2
⎡
⎣gij( x ) ⎤
⎦ = ⎢ 0 a ( cosh x −cos x ) 0 ⎥
⎢
⎥
2 2 1 2 2
⎢ 0 0 a sinh x sin x ⎥
⎣
⎦
44.7 Elliptical cylindrical coordinates
coordinate system by
1 2 3
( x , x , x ) are defined relative to a rectangular Cartesian
z = acosh
x cos x
1 1 2
z = asinh
x sin x
z
2 1 2
= x
3 3
1 2 3
1
where a > 0 . How must ( x , x , x ) be restricted so as to make ẑ xˆ
− one-to-one?
Discuss the coordinate curves and coordinate surfaces. Also, show that
( + )
2 2 1 2 2
⎡a sinh x sin x
0 0⎤
⎢
⎥
2 2 1 2 2
⎡
⎣gij( x ) ⎤
⎦ = ⎢ 0 a ( sinh x + sin x ) 0⎥
⎢
⎥
⎢ 0 0 1⎥
⎣
⎦
44.8 For the cylindrical coordinate system show that
g = (cos x ) i + (sin x ) i
2 2
1 1 2
g =− x (sin x ) i + x (cos x ) i
g
1 2 1 2
2 1 2
= i
3 3
44.9 At a point x in E , the components of the position vector rx ( ) = x−
0 E
with respect to the
basis { i ,..., 1
i
N } associated with a rectangular Cartesian coordinate system are
1 N
z ,..., z .
This observation follows, of course, from (44.16). Compute the components of rx ( ) with
respect to the basis { g1( x), g2( x), g3( x )}
for (a) cylindrical coordinates, (b) spherical
coordinates, and (c) parabolic coordinates. You should find that