Mathematical Methods for Physics and Engineering : A Comprehensive Guide

VECTOR CALCULUS

10.21 Paraboloidal coordinatesu, v, φare defined in terms of Cartesian coordinates by

x=uvcosφ, y=uvsinφ, z=^12 (u^2 −v^2 ).

Identify the coordinate surfaces in theu, v, φsystem. Verify that each coordinate

surface (u= constant, say) intersects every coordinate surface on which one of

the other two coordinates (v, say) is constant. Show further that the system of

coordinates is an orthogonal one and determine its scale factors. Prove that the

u-component of∇×ais given by

1

(u^2 +v^2 )^1 /^2

(

aφ

v

+

∂aφ

∂v

)

−

1

uv

∂av

∂φ

.

10.22 Non-orthogonal curvilinear coordinates are difficult to work with and should be
avoided if at all possible, but the following example is provided to illustrate the
content of section 10.10.
In a new coordinate system for the region of space in which the Cartesian
coordinatezsatisfiesz≥0, the position of a pointris given by (α 1 ,α 2 ,R), where
α 1 andα 2 are respectively the cosines of the angles made byrwith thex-andy-
coordinate axes of a Cartesian system andR=|r|. The ranges are− 1 ≤αi≤1,
0 ≤R<∞.
(a) Expressrin terms ofα 1 ,α 2 ,Rand the unit Cartesian vectorsi,j,k.
(b) Obtain expressions for the vectorsei(=∂r/∂α 1 ,...) and hence show that the
scale factorshiare given by

h 1 =

R(1−α^22 )^1 /^2

(1−α^21 −α^22 )^1 /^2

,h 2 =

R(1−α^21 )^1 /^2

(1−α^21 −α^22 )^1 /^2

,h 3 =1.

(c) Verify formally that the system is not an orthogonal one.

(d) Show that the volume element of the coordinate system is

dV=

R^2 dα 1 dα 2 dR

(1−α^21 −α^22 )^1 /^2

,

and demonstrate that this is always less than or equal to the corresponding

expression for an orthogonal curvilinear system.

(e) Calculate the expression for (ds)^2 for the system, and show that it differs

from that for the corresponding orthogonal system by

2 α 1 α 2 R^2

1 −α^21 −α^22

dα 1 dα 2.

10.23 Hyperbolic coordinatesu, v, φare defined in terms of Cartesian coordinates by

x=coshucosvcosφ, y=coshucosvsinφ, z=sinhusinv.
Sketch the coordinate curves in theφ= 0 plane, showing that far from the origin
they become concentric circles and radial lines. In particular, identify the curves
u=0,v=0,v=π/2andv=π. Calculate the tangent vectors at a general
point, show that they are mutually orthogonal and deduce that the appropriate
scale factors are
hu=hv=(cosh^2 u−cos^2 v)^1 /^2 ,hφ=coshucosv.
Find the most general functionψ(u)ofuonly that satisfies Laplace’s equation
∇^2 ψ=0.
10.24 In a Cartesian system,AandBare the points (0, 0 ,−1) and (0, 0 ,1) respectively.
In a new coordinate system a general pointPis given by (u 1 ,u 2 ,u 3 )with
u 1 =^12 (r 1 +r 2 ),u 2 =^12 (r 1 −r 2 ),u 3 =φ;herer 1 andr 2 are the distancesAPand
BPandφis the angle between the planeABPandy=0.