2.1 Cartesian, cylindrical, and spherical coordinates

Coordinate systems are the backbone of spatial understanding in physics and math. They help us pinpoint locations and describe motion in space. Cartesian, cylindrical, and spherical coordinates each offer unique ways to represent points.

These systems have different strengths. Cartesian is great for simple 3D space, cylindrical works well for objects with circular symmetry, and spherical is perfect for describing things like planetary orbits or electromagnetic fields.

Coordinate Systems

Cartesian Coordinates

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• Rectangular coordinate system consisting of three mutually perpendicular axes (x, y, and z)
• Each point in space is represented by an ordered triple (x, y, z)
• Useful for describing positions and vectors in three-dimensional space
• Commonly used in physics, engineering, and mathematics

Cylindrical Coordinates

• Three-dimensional coordinate system consisting of radial distance $\rho$, azimuthal angle $\phi$, and height $z$
• Radial distance $\rho$ measures the distance from the origin in the xy-plane
• Azimuthal angle $\phi$ measures the angle in the xy-plane from the positive x-axis (counterclockwise)
• Height $z$ measures the distance along the z-axis, perpendicular to the xy-plane
• Useful for describing systems with cylindrical symmetry (pipes, cylinders, or objects with rotational symmetry)

Spherical Coordinates

• Three-dimensional coordinate system consisting of radial distance $[r](https://www.fiveableKeyTerm:r)$, polar angle $\theta$, and azimuthal angle $\phi$
• Radial distance $r$ measures the distance from the origin to the point in space
• Polar angle $\theta$ measures the angle from the positive z-axis to the point ($0 \leq \theta \leq \pi$)
• Azimuthal angle $\phi$ measures the angle in the xy-plane from the positive x-axis (counterclockwise) ($0 \leq \phi < 2\pi$)
• Useful for describing systems with spherical symmetry (gravitational fields, electromagnetic waves, or atoms)

Components of Coordinate Systems

Origin and Axes

• Origin is the point where all coordinates are zero (0, 0, 0)
• Axes are the reference lines along which coordinates are measured
  • In Cartesian coordinates: x-axis, y-axis, and z-axis
  • In cylindrical coordinates: radial axis ($\rho$), azimuthal axis ($\phi$), and z-axis
  • In spherical coordinates: radial axis ($r$), polar axis ($\theta$), and azimuthal axis ($\phi$)

Radial Distance

• Measures the distance from the origin to a point in space
• Denoted by $\rho$ in cylindrical coordinates and $r$ in spherical coordinates
• Always non-negative ($\rho \geq 0$ and $r \geq 0$)

Angular Measurements

• Azimuthal angle $\phi$:
  • Measures the angle in the xy-plane from the positive x-axis (counterclockwise)
  • Range: $0 \leq \phi < 2\pi$ in both cylindrical and spherical coordinates
• Polar angle $\theta$ (spherical coordinates only):
  • Measures the angle from the positive z-axis to the point
  • Range: $0 \leq \theta \leq \pi$

Conventions

Right-Handed System

• Standard convention for orienting coordinate systems in three-dimensional space
• Determined by the direction of rotation from the positive x-axis to the positive y-axis (counterclockwise)
• The positive z-axis points in the direction of the thumb when the fingers of the right hand curl from the positive x-axis to the positive y-axis
• Ensures consistency in vector operations and cross products
• Commonly used in physics, engineering, and mathematics