Coordinate Conversion in Calculus

Flashcards covering key concepts related to coordinate conversion in calculus, including definitions and formulas.

Rectangular Coordinates

Coordinates in the standard form (x, y, z) used in 3D space.

Cylindrical Coordinates

Coordinates defined by (r, θ, z), where r is the radius, θ is the angle, and z is the height.

Spherical Coordinates

Coordinates given in the form (ρ, θ, φ), where ρ is the distance from the origin, θ is the azimuthal angle, and φ is the polar angle.

Conversion from Rectangular to Cylindrical

To convert, use x= r cos(θ), y= r sin(θ), and z= z.

Conversion from Rectangular to Spherical

The transformation is given by ρ = √(x² + y² + z²), θ = tan⁻¹(y/x), φ = cos⁻¹(z/ρ).

Volume Element in Rectangular Coordinates (dv)

The volume element in rectangular coordinates is dv = dx dy dz.

Volume Element in Cylindrical Coordinates

In cylindrical coordinates, dv = r dr dθ dz.

Volume Element in Spherical Coordinates

In spherical coordinates, the volume element is dv = ρ² sin(φ) dρ dφ dθ.