Polar-Cylindrical Coordinates Calculator
Understanding Coordinate Systems
Coordinate System Relationships
Polar and cylindrical coordinates are related systems for describing points in 2D and 3D space.
Polar to Cylindrical:
- ρ = r
- φ = θ
- z = 0 (by default)
Cylindrical to Polar:
- r = √(ρ² + z²)
- θ = φ
Cartesian Relations:
- x = ρ cos(φ)
- y = ρ sin(φ)
- z = z
Applications and Properties
- Physical Applications:
- Electromagnetic Fields
- Fluid Dynamics
- Heat Conduction
- Wave Propagation
- Mathematical Properties:
- Rotational Symmetry
- Periodic Functions
- Vector Fields
- Surface Integrals
- Special Cases:
- Axis Points (ρ = 0)
- Planar Sections (z = constant)
- Radial Lines (φ = constant)
- Cylindrical Shells (ρ = constant)
Advanced Concepts
Volume Element
dV = ρ dρ dφ dz
Surface Element
dS = ρ dφ dz
Scale Factors
hρ = 1, hφ = ρ, hz = 1
Gradient
∇ in cylindrical coordinates