To compare and contrast: the rectangular, cylindrical and spherical coordinates systems.

Tazmin Horton 2020-12-28 Answered
To compare and contrast: the rectangular, cylindrical and spherical coordinates systems.
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Expert Answer

Delorenzoz
Answered 2020-12-29 Author has 91 answers
Given:
Point is represented as (r,θ,z).
Proof:
A point in cylindrical coordinate is represented by (r,θ.z).
(r,θ) are polar coordinates
z is the usual - coordinates
The rectangular (x, y, z) and the spherical coordinates (r,θ,z) are related as
x=rcosθ
y=rsinθ
z=z
These are used to convert from cylindrical to rectangular coordinates.
Rectangular to cylindrical coordinates:
r2=x2+y2
tanθ=yx
z=z
A point in spherical coordinate is represented by (ρ,θ,φ).
The rectangular (x, y, z) and the spherical coordinates (ρ,θ,φ) are related as
x=ρsinφcosθ
y=ρsinφsinθ
z=ρcosφ
These are used to convert from spherical to rectangular coordinates.
Rectangular to spherical coordinates:
ρ2=x2+y2+z2
tanθ=yx
φ=cos1(zx2+y2+z2)
Spherical to rectangular coordinates:
r=ρsinφ
θ=φ
z=ρcosφ
Cylindricalto spherical coordinates:
ρ2=r2+z2
θ=θ
φ=cos1(zr2+z2)
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