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| In 3 dimensions, the most common system is '''rectangular''' coordinates (i.e., ''(x,y,z)''). Others include '''cylindrical''' coordinates (i.e., ''(r,θ,z)'') and '''spherical''' coordinates (i.e., ''(ρ,θ,φ)'' where ''ρ'' is the radial distance, ''θ'' is the '''polar angle''', and ''φ'' is the '''azimuthal angle'''). Polar angles are measured in comparison to the polar axis, and range from 0 to ''π'' (radians). Azimuthal angles are measured as rotation around the polar axis. | In 3 dimensions, the most common system is '''rectangular''' coordinates (i.e., ''(x,y,z)''). Others include '''cylindrical''' coordinates (i.e., ''(r,θ,z)'') and '''spherical''' coordinates (i.e., ''(ρ,θ,φ)'' where ''ρ'' is the radial distance, ''θ'' is the '''azimuthal angle''', and ''φ'' is the '''polar angle'''). Azimuthal angles are measured as rotation around the polar axis. Polar angles are measured in comparison to the polar axis, and range from 0 to ''π'' (radians). Note that in some contexts, the roles of ''θ'' and ''φ'' are reversed. |
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| Note that in some texts, ''θ'' and ''φ'' are reversed. Here ''φ'' is the polar angle and ''θ'' is the azimuthal angle. |
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Coordinate System
A coordinate system is a mapping of parameters to locations.
Contents
Description
A coordinate system maps a set of coordinate parameters to a unique location.
In 2 dimensions, the most common system is Cartesian coordinates (i.e., (x,y)). Beyond this, there is also polar coordinates (i.e., (r,θ) where r is a radius and θ is an angle).
In 3 dimensions, the most common system is rectangular coordinates (i.e., (x,y,z)). Others include cylindrical coordinates (i.e., (r,θ,z)) and spherical coordinates (i.e., (ρ,θ,φ) where ρ is the radial distance, θ is the azimuthal angle, and φ is the polar angle). Azimuthal angles are measured as rotation around the polar axis. Polar angles are measured in comparison to the polar axis, and range from 0 to π (radians). Note that in some contexts, the roles of θ and φ are reversed.
Changing System
Polar to Cartesian
x = r cos(θ)
y = r sin(θ)
The Jacobian is given by J = r, as in dA = dxdy = rdrdθ
Cartesian to Polar
tan(θ) = y/x or θ = tan-1(y/x)
r2 = x2 + y2 or r = √(x2 + y2)
J = 1/r, as in dA = drdθ = (1/r)dxdy
Cylindrical to Rectangular
x = r cos(θ)
y = r sin(θ)
z = z
J = r, as in dV = dxdydz = rdrdθdz
Rectangular to Cylindrical
tan(θ) = y/x or θ = tan-1(y/x)
r2 = x2 + y2 or r = √(x2 + y2)
z = z
J = 1/r, as in dV = drdθdz = (1/r)dxdydz
Spherical to Rectangular
Note that in some texts, θ and φ are reversed. Here φ is the polar angle and θ is the azimuthal angle.
x = ρ sin(φ) cos(θ)
y = ρ sin(φ) sin(θ)
z = ρ cos(φ)
J = ρ2 sin(φ), as in dV = dxdydz = ρ2 sin(φ) dρdθdφ
Rectangular to Spherical
ρ2 = x2 + y2 + z2 or ρ = √(x2 + y2 + z2)
tan(θ) = y/x or θ = tan-1(y/x)
cos(φ) = z/ρ = z/√(x2 + y2 + z2) or φ = cos-1(z/√(x2 + y2 + z2))
Cylindrical to Spherical
Note that in some texts, θ and φ are reversed. Here φ is the polar angle and θ is the azimuthal angle.
ρ2 = r2 + z2 or ρ = √(r2 + z2)
θ = θ
cos(φ) = z/ρ = z/√(r2 + z2) or φ = cos-1(z/√(r2 + z2))
Spherical to Cylindrical
r = ρ sin(φ)
θ = θ
z = ρ cos(φ)