Coordinate System

A coordinate system is a mapping of parameters to locations.

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)

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