Before proceeding any further, it is useful to introduce a new coordinate system that is particularly suited to the problems we shall encounter.
In two dimensions, the normal Cartesian system sets an origin and then specifies the position of a point as its distance from this origin in two perpendicular directions, the x, and y directions. In the polar coordinate system, the two coordinates used are the straight line distance of the point from the origin, R, and the angle Φ that a line from the point to the origin makes with a specified axis (normally chosen as the equivalent of the x axis in the Cartesian system):
The above diagrams represent the two types of coordinate system being applied to the same point. The coordinates of the point are (x,y) or (r,Φ) respectively.
From the above diagram, it should be clear (from basic trigonometry) that there are definite relations between the coordinates in the two systems:
These relations may be used to be substitute for x and y in any expression to convert it into polar coordinates. To illustrate that polar coordinates can greatly simplify rotational problems, we will consider the equation of a circle of radius a in both coordinate systems:
Clearly, the second equation is much easier to manipulate than the first.
Extension of the coordinate systems to three dimensions is relatively simple. In the Cartesian case, we add another axis perpendicular to the x and y axes and call it the z axis, the third coordinate is the distance from the point to the origin parallel to this axis. In polar coordinates the third coordinate is the angle θ between the z axis and a line from the point to the origin.
In this case, the relations between the Cartesian and polar coordinates are:
It is appropriate to note here that in three dimensional problems, it is common to encounter the sum of the second derivatives with respect to each Cartesian variable, and this quantity is abbreviated to the following symbol:
The symbol is a convenient abbreviation, and is known as the laplacian. It is read as “del squared”. In polar coordinates, the laplacian is as follows:
where Λ2 is the legendrian, given by:
Though these expressions may look exceedingly complex, the mathematics required to apply them is generally basic differentiation, requiring a knowledge of only the product and quotient rules. The mathematics involved can be somewhat long and tedious, but fortunately there are some simplifications that may be made in the cases we shall consider.