Mathematics for Physics I - Coordinate Systems

Cartesian Coordinates

The Cartesian coordinate system is the most familiar way to specify positions in space. In two dimensions, we use ordered pairs (x, y) to represent positions relative to perpendicular axes.

Two-Dimensional Cartesian Coordinates

A point P in the xy-plane is represented by the ordered pair (x, y), where:

x is the horizontal distance from the y-axis
y is the vertical distance from the x-axis

The distance between two points \(P_1(x_1, y_1)\) and \(P_2(x_2, y_2)\) is given by:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Three-Dimensional Cartesian Coordinates

A point P in three-dimensional space is represented by the ordered triple (x, y, z), where:

x is the distance from the yz-plane
y is the distance from the xz-plane
z is the distance from the xy-plane

The distance between two points \(P_1(x_1, y_1, z_1)\) and \(P_2(x_2, y_2, z_2)\) is:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

Interactive visualization of Cartesian coordinates in two and three dimensions

Polar Coordinates

Polar coordinates provide an alternative way to specify positions in a plane, particularly useful for problems with circular symmetry.

Definition

A point P in the plane is represented by the ordered pair (r, θ), where:

r is the distance from the origin to the point
θ is the angle measured counterclockwise from the positive x-axis

The relationship between Cartesian and polar coordinates is:

\[ x = r\cos\theta \] \[ y = r\sin\theta \]

And conversely:

\[ r = \sqrt{x^2 + y^2} \] \[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \]

Applications

Polar coordinates are particularly useful for:

Describing circular motion
Analyzing systems with radial symmetry
Simplifying certain integrals
Representing waves and oscillations

The area element in polar coordinates is:

\[ dA = r \, dr \, d\theta \]

The gradient operator in polar coordinates is:

\[ \nabla f = \frac{\partial f}{\partial r}\hat{r} + \frac{1}{r}\frac{\partial f}{\partial \theta}\hat{\theta} \]

Visualization of polar coordinates and their relationship to Cartesian coordinates

Cylindrical and Spherical Coordinates

Cylindrical Coordinates

Cylindrical coordinates (ρ, φ, z) are an extension of polar coordinates to three dimensions:

ρ is the distance from the z-axis
φ is the angle in the xy-plane from the positive x-axis
z is the height above the xy-plane

Conversion to Cartesian coordinates:

\[ x = \rho\cos\phi \] \[ y = \rho\sin\phi \] \[ z = z \]

The volume element in cylindrical coordinates is:

\[ dV = \rho \, d\rho \, d\phi \, dz \]

Spherical Coordinates

Spherical coordinates (r, θ, φ) are particularly useful for problems with spherical symmetry:

r is the distance from the origin
θ is the polar angle from the positive z-axis
φ is the azimuthal angle in the xy-plane from the positive x-axis

Conversion to Cartesian coordinates:

\[ x = r\sin\theta\cos\phi \] \[ y = r\sin\theta\sin\phi \] \[ z = r\cos\theta \]

The volume element in spherical coordinates is:

\[ dV = r^2\sin\theta \, dr \, d\theta \, d\phi \]

Visualization of cylindrical and spherical coordinate systems

Applications in Physics

Rotational Motion

Polar and cylindrical coordinates naturally describe rotational motion, such as:

Planetary orbits
Rotating machinery
Angular momentum

Example: The centripetal acceleration in polar coordinates is simply \(a_r = -r\omega^2\).

Electromagnetic Fields

Different coordinate systems simplify electromagnetic problems:

Cylindrical coordinates for wire and solenoid fields
Spherical coordinates for point charges and dipoles

Example: The electric field of a point charge has only a radial component in spherical coordinates: \(E_r = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\).

Wave Phenomena

Different coordinate systems simplify wave equations:

Cartesian coordinates for plane waves
Cylindrical coordinates for waves in cylindrical cavities
Spherical coordinates for spherical waves

Example: The wave equation in spherical coordinates allows separation of variables for spherically symmetric problems.

Quantum Mechanics

Coordinate systems are crucial in quantum mechanics:

Hydrogen atom wavefunctions are naturally expressed in spherical coordinates
Cylindrical coordinates for cylindrically symmetric potentials

Example: The hydrogen atom wavefunctions are products of radial functions and spherical harmonics.

Choosing the Right Coordinate System

Selecting the appropriate coordinate system can dramatically simplify mathematical analysis. Here are some guidelines:

Use Cartesian Coordinates When:

The problem involves straight lines or planes
The boundary conditions are aligned with the coordinate axes
The problem has translational symmetry
Working with rectangular regions

Use Polar/Cylindrical Coordinates When:

The problem involves circles or cylinders
There is rotational symmetry around an axis
Working with circular or cylindrical regions
Analyzing rotational motion

Use Spherical Coordinates When:

The problem involves spheres
There is spherical symmetry
Working with radial fields (gravitational, electric)
Analyzing problems with point sources

General Principle:

Choose the coordinate system that:

Matches the symmetry of the problem
Simplifies the boundary conditions
Reduces the number of variables needed
Makes the equations separable when possible

Key Insight:

The laws of physics are invariant under coordinate transformations. The physical reality doesn't change when we switch coordinate systems—only our mathematical description of it. Choosing the right coordinate system doesn't change the physics, but it can make the mathematics much more tractable.

Graphs and Coordinate Systems