Mathematical Handbook by Sotirios Persidis

Author:Sotirios Persidis [Persidis, Sotirios]
Language: eng
Format: mobi, epub
ISBN: 9789607610126
Amazon: 9607610121
Publisher: ESPI PUBLISHING LTD
Published: 2007-09-15T00:00:00+00:00

h = h = 1, h = h = ρ, h = h = 1

1

ρ

2

φ

3

z

e = e , e = e , e = e , A = A e + A e + A e 1

ρ

2

φ

3

z

ρ ρ

φ φ

z z

dr = dρe + ρdφe + dze

ρ

φ

z

ds 2 = dρ 2 + ρ 2 dφ 2 + dz 2

Fig. 13-2

dV = ρdρdφdz

d S = ρdφdze + dρdze + ρdρdφe

ρ

φ

z

\DΦ=(\dΦ/\dρ)*e_ρ+(1/ρ)*(\

\D\*A=(1/ρ)*(\d(ρ*A_ρ)/\dρ)+(1/

dΦ/\dφ)*e_φ+(\dΦ/\dz)*e_z

ρ)*(\dA_φ/\dφ)+\dA_z/\dz

\D\xA=(1/ρ)*(\dA_z/\dφ-\d(ρ*A_φ)/\dz)*e_ρ+(\dA_ρ/\dz-\dA_z/\dρ)*e_

θ+(1/ρ)*(\d(ρ*A_φ)/\dρ-\dA_ρ/\dφ)*e_φ

CHAP. 13

CURVILINEAR COORDINATES

191

(A\*\D){Φ}=A_ρ*\dΦ/\dρ+(A_φ/ρ)*\dΦ/\dφ+A_z*\

dΦ/\dz

(\D)^2{Φ}=(\d/\dρ)^2{Φ}+(1/ρ)*\dΦ/\dρ+(1/

Ext

ρ^2)*(\d/\dφ)^2{Φ}+(\d/\dz)^2{Φ}

Spherical coordinates ( r, θ, φ)

x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ

r = ( x 2 + y 2 + z 2)1/2, θ = cos−1( z/ r),

φ=cos^(-1)(x/(x^2+y^2)^(1/2))=sin^(-

1)(y/(x^2+y^2)^(1/2))

with 0 ≤ r < ∞, 0 ≤ θ ≤ π, 0 ≤ φ < 2 π.

Spherical coordinates

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