The Chemistry Maths Book, Second Edition

10 Functions in 3 dimensions

10.1 Concepts

A function of three variables,f(x, y, z), in which the variables are the coordinates of a

point in ordinary three-dimensional space is called a function of position, a point

function, or a field. A great variety of physical quantities are described by such

functions. For example, the distribution of mass in a physical body is described by the

(volume) mass density, a function of position, the temperature at each point in a body

defines a temperature field, and the velocity at each point in a mass of moving fluid is

a vector function of position, a velocity field (see Chapter 16). Electric and magnetic

fields are vector functions of position, atomic and molecular wave functions are scalar

functions of position.

When a function of position is expressed in the formf(x, y, z)it is implied that

it is a function of the cartesian coordinates of a point.

1

The value of a function at a

given point cannot however depend on the particular system of coordinates used

to specify its position. We saw in Chapter 9 that, for points in a plane, a function and

the region in which it is defined may sometimes (in fact, often) be expressed more

simply in terms of coordinates other than cartesian. The most important and most

widely used coordinates, other than the cartesian, are the spherical polar coordinates.

They are presented in Section 10.2. Functions of position are discussed in Section 10.3

and volume integrals in 10.4. The Laplacian operator, so important in the physical

sciences, is discussed in Section 10.5. In these, the discussion is restricted to the

cartesian and spherical polar coordinate systems. The general coordinate system

is the subject of Section 10.6. The use of vectors and matrices for the description

of systems and processes in three dimensions is discussed in Chapters 16, 18,

and 19.

10.2 Spherical polar coordinates

The position of a point in a three-dimensional space is specified uniquely by its three

coordinates in a given coordinate system. In the cartesian system shown in Figure 10.1,

1

The analytical geometry of three dimensions has its origins in Clairaut’s Recherchesof 1731, in which he gave

x

2

1 + 1 y

2

1 + 1 z

2

1 = 1 a

2

as the equation of a sphere of radius a, and in the second volume of Euler’s Introductioof 1748.

The systematic theory was developed by Monge in papers from 1771 and in two influential textbooks written

for his students at the École Polytechnique. Gaspar Monge (1746–1818), the son of Jacques Monge, peddler,

knife-grinder, and respecter of education, developed a ‘descriptive geometry’ that formed the basis for modern

engineering drawing; the method was classified for a time as a military secret, and published in 1799 in the

textbook Géométrie descriptive. Monge was the first Director of the École Polytechnique formed in 1794 by the

National Convention of the Revolution for the training of engineers and scientists (who show ‘a constant love of

liberty, equality, and a hatred of tyrants’). The École became a model for colleges throughout Europe and the

United States. Teachers at the École included Laplace, Lagrange, and Sylvestre François Lacroix (1765–1843),

whose textbook on the calculus was translated into English in 1816 and was influential (with other texts from the

École) in bringing European methods to England and the United States. In the textbook Application de l’analyse

à la géométrie, 1807, Monge discussed the analytical geometry of two and three dimensions. He showed how the

coordinates of a point are determined by the perpendiculars from three coordinate planes.