Coordinate Systems

Home Physics Equations Classical Mechanics Coordinate Systems


The coordinate system is illustrated in Figure below. The location of a pointin three dimensional space may be specified by an ordered set of numbers(x, y, z). The ranges for the coordinate parameters are:





The unit vectors unit vectors, unit vectors, and unit vectors are also illustrated in Figure
The rectangular coordinate system.

These unit vectors are every where mutually orthogonal. The "del" operator in rectangular coordinates is simply:
del operator in rectangular coordinates
The Laplacian operator in rectangular coordinates is :
Laplacian operator in rectangular coordinates

Cylindrical Coordinates

The coordinate system is illustrated in Figure below. The location of a pointin three dimensional space may be specified by an ordered set of numbers(r, theta, z). The ranges for the coordinate parameters are:




The cylindrical coordinate system.

The relationship between rectangular and cylindrical coordinates is summarized as follows:


z = z       z = z

The unit vectors unit vector r, unit vector theta, and unit vector z are also illustrated in above figure. These unit vectors are every where mutually orthogonal. In contrast torectangular coordinates, the unit vectors unit vector r and unit vector theta change direction dependingon the particular point in space. For this reason, it is critical to take care when executing diferential operations in cylindrical coordinates.
For example, and . The "del" operator in cylindrical coordinates is:
del operator in cylindrical coordinates

The Laplacian operator in cylindrical coordinates is :
Laplacian operator in cylindrical coordinates


Spherical Coordinates

The coordinate system is illustrated in Figure below. The location of a pointin three dimensional space may be speci´┐Żed by an ordered set of numbers. The ranges for the coordinate parameters are:





Note carefully that the definitions of theta are very diferent in the cylindricaland spherical coordinate systems! The relationship between rectangular and spherical coordinates is summarized as follows:

The unit vectors , and are also illustrated in Figure below.
These unit vectors are every where mutually orthogonal. In contrast to rectangular coordinates, each of these unit vectors changes direction dependingon the particular point in space. For this reason, it is critical to take care when executing diferential operations in spherical coordinates.
For example, and The "del" operator in spherical coordinates is:
del operator in spherical coordinates
The Laplacian operator in spherical coordinates is :
Laplacian operator in spherical coordinates

The spherical coordinate system