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Handbook of Linear Partial Differential Equations for Engineers and Scientists, 2nd Edition
book

Handbook of Linear Partial Differential Equations for Engineers and Scientists, 2nd Edition

by Andrei D. Polyanin, Vladimir E. Nazaikinskii
December 2015
Intermediate to advanced content levelIntermediate to advanced
1643 pages
59h 33m
English
Chapman and Hall/CRC
Content preview from Handbook of Linear Partial Differential Equations for Engineers and Scientists, 2nd Edition
29.3. Other Special Orthogonal Coordinates 1505
Components of the metric tensor:
g
σσ
= a
2
σ
2
τ
2
σ
2
1
, g
ττ
= a
2
σ
2
τ
2
1 τ
2
, g
zz
= 1.
2
. Special coordinate system u, v, z:
σ = cosh u, τ = cos v, z = z;
x = a cosh u cos v, y = a sinh u sin v, z = z
(0 u < , 0 v π, −∞ < z < ).
Components of the metric tensor:
g
uu
= g
vv
= a
2
(sinh
2
u + sin
2
v), g
zz
= 1.
3
. Laplacian:
f =
1
a
2
(sinh
2
u+sin
2
v)
2
f
∂u
2
+
2
f
∂v
2
+
2
f
∂z
2
=
σ
2
1
a
2
(σ
2
τ
2
)
∂σ
p
σ
2
1
∂f
∂σ
+
1τ
2
a
2
(σ
2
τ
2
)
∂τ
p
1τ
2
∂f
∂τ
+
2
f
∂z
2
.
Remark 29.4. The elliptic cylinder coordinates σ, τ are also used a s elliptic coordinates on the
plane xy.
29.3.4 Conical Coordinates
Transformations of coordinates:
x = ±
uvw
ab
, y
2
=
u
2
(v
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Publisher Resources

ISBN: 9781466581494