The Chemistry Maths Book, Second Edition

10.3 Functions of position 297

The temperature field is therefore described equally well by the function

g(r, 1 θ, 1 φ) 1 = 1 r

2

(cos

2

1 θ 1 − 1 sin

2

1 θ)such that, at any point in V,

T 1 = 1 f(x, y, z) 1 = 1 g(r, θ, φ)

In this example, the transformation from cartesian to spherical polar coordinates has

led to two simplifications of the representation of the field. Firstly, whereas Tdepends

on all three cartesian coordinates, x, y, and z, it only depends on two of the spherical

polar coordinates, rand θ, that is, the field is independent of the value of the angle

φ, and is therefore cylindrically symmetric about the polar (z) axis. The second

simplification is that the variables in the function ghave been separated; that is, the

functiong(r, 1 θ)has been factorized as the product of a function of rand a function

of θ. Such simplifications are important for the evaluation of multiple integrals and

for the solution of partial differential equations.

EXAMPLE 10.3Express in spherical polar coordinates: (i) (x

2

1 + 1 y

2

1 + 1 z

2

)

122

,

(ii) , (iii)

(i)(x

2

1 + 1 y

2

1 + 1 z

2

)

122

1 = 1 r,

(ii)

(iii)

0 Exercises 10–13

Density functions

Consider the distribution of mass in a three-dimensional body (see Section 5.6 for

a linear distribution of mass). Let P be some point within the body, and let ∆mbe

the mass in a volume∆vsurrounding the point P, as

shown in Figure 10.4. The ratio∆m 2 ∆vis then the mass

per unit volume, or the average mass density, in∆v.

If the mass is not distributed uniformly throughout

the body, the value of this ratio depends not only on

the position of the volume∆vbut also on the size and

shape of∆v. As the size of the volume is reduced to

zero,∆v 1 → 10 , the ratio approaches a limit

ρ= (10.3)













→

lim

∆

∆

v 0 ∆v

m

∂

∂

++ =

∂

∂

=

r

xyz

r

()r

22212

1

∂

∂

++ = ++ ×==

−

x

xyz xyz x

x

r

()() sinc

22212 22212

1

2

2 θ oosφ

∂

∂

++

r

()xyz

22212

∂

∂

++

x

()xyz

22212

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Figure 10.4