Page 366 - Intro to Tensor Calculus

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2
3
1
11. Toroidal coordinates (u, v, φ)= (x ,x ,x )
2 2
x = a sinh v cos φ , 0 ≤ u< 2π h = h 2
1
2
cosh v − cos u
2 a
h =
2
y = a sinh v sin φ , −∞ <v < ∞ (cosh v − cos u) 2
2
cosh v − cos u 2
2 a sinh v
z = a sin u h = 2
3
(cosh v − cos u)
0 ≤ φ< 2π
,
cosh v − cos u
The coordinate curves are formed by the intersection of the coordinate surfaces
2 2

2 2 a cos u a
x + y + z − = Spheres
2 ,
sin u sin u
2
2
p
2 2 cosh v 2 a
x + y − a + z = 2 , Tores
sinh v
sinh v
planes
y = x tan φ,

1 sin u 2 sinh v (cos u coshv − 1)
= = −
11 cos u − cosh v 33 cos u − cosh v

2 sinh v 1 sinh v
= =
22 cos u − cosh v 12 cos u − cosh v

1 sin u 2 sin u
= =
22 − cosu +cosh v 21 cos u − cosh v
2
1 sin usinh v 3 sin u
= =
33 − cosu +cosh v 31 cos u − cosh v

2 sinh v 3 cos u cosh v − 1
= =
11 − cosu +cosh v 32 cos u sinh v − cosh v sinh v