Page 443 - Advanced engineering mathematics

12.10 Curvilinear Coordinates 423

The Laplacian in various coordinate systems is often encountered in connection with
diffusion problems, wave motion and potential theory.

SECTION 12.10 PROBLEMS

1. Compute the scale factors for cylindrical coordinates. (a) Sketch the coordinate surfaces u = constant, v =
Use them to compute ∇· F and ∇× F if F(r,θ, z) is a constant, and z = constant. Are these coordinates
vector field in cylindrical coordinates. If g(r,θ, z) is a orthogonal?
2
scalar field, compute ∇g and ∇ g.
(b) Determine the scale factors h u , h v , h z .
2. Elliptic cylindrical coordinates are defined by
(c) Determine ∇ f (u,v, z) in this system.
x = a cosh(u)cos(v), y = a sinh(u)sin(v), z = z, (d) Determine ∇· F(u,v, z) and ∇× F(u,v, z) in this
system.
where u ≥0, 0≤v<2π and z can be any real number.
2
(e) Determine ∇ f (u,v, z).
(a) Sketch the coordinate surfaces u = constant, v =
4. Parabolic cylindrical coordinates are defined by
constant, and z = constant.
1
2
2
(b) Determine the scale factors h u , h v , h z . x = uv, y = (u − v ), z = z,
2
(c) Determine ∇ f (u,v, z) in this system.
with u ≥ 0and v and z any real numbers.
(d) Determine ∇· F(u,v, z) and ∇× F(u,v, z) in this
system. (a) Sketch the coordinate surfaces u = constant, v =
constant, and z = constant.
2
(e) Determine ∇ f (u,v, z).
(b) Determine the scale factors h u , h v , h z .
3. Bipolar coordinates are defined by
(c) Determine ∇ f (u,v, z) in this system.
a sinh(v) a sin(u) (d) Determine ∇· F(u,v, z) and ∇× F(u,v, z) in this
x = , y = , z = z,
cosh(v) − cos(u) cosh(v) − cos(u) system.
2
with u and z any real numbers and 0 ≤ v< 2π. (e) Determine ∇ f (u,v, z).

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October 14, 2010 14:53 THM/NEIL Page-423 27410_12_ch12_p367-424