The Chemistry Maths Book, Second Edition

296 Chapter 10Functions in 3 dimensions

The conversion from cartesian coordinates to spherical polar coordinates makes use

of the following relations:

(10.2)

in which the inverse functions have their principal values (see Section 3.5).

EXAMPLE 10.2Find the spherical polar coordinates of the point(x, y, z) 1 = 1 (−1, 2, −3).

r

2

1 = 1 x

2

1 + 1 y

2

1 + 1 z

2

1 = 1 14, 1 r 1 = 1

θ 1 = 1 cos

− 1

(z 2 r) 1 = 11 ≈ 1 143.3°

φ 1 = 1 tan

− 1

(y 2 x) 1 + 1 π 1 = 1 tan

− 1

(−2) 1 + 1 π 1 ≈ 1 116.6°

0 Exercises 4–9

10.3 Functions of position

A function of position, or field, is a function of the three coordinates within some

region of three-dimensional space. Let the region V(for volume) in Figure 10.3

represent, for example, a body with non-uniform temperature; the temperature is

a function of position,

T 1 = 1 f(x, 1 y, 1 z)

Then, if the cartesian coordinates of the point Pare

(x

p

, y

p

, z

p

), the temperature at this point is

T

p

1 = 1 f(x

p

, y

p

, z

p

)

For example, iff(x, 1 y, 1 z) 1 = 1 z

2

1 − 1 x

2

1 − 1 y

2

then

T

p

1 = 1 z

2

p

1 − 1 x

2

p

1 − 1 y

2

p

The temperature at a point cannot depend on the particular system of coordinates

used to specify the position of the point. If the spherical coordinates at P are (r

p

, 1 θ

p

, 1 φ

p

)

then, by equations (10.1),

T

p

1 = 1 r

p

2

1 cos

2

1 θ

p

1 − 1 r

p

2

1 sin

2

1 θ

p

1 cos

2

1 φ

p

1 − 1 r

p

2

1 sin

2

1 θ

p

1 sin

2

1 φ

p

1 = 1 r

p

2

(cos

2

1 θ

p

1 − 1 sin

2

1 θ

p

)

cos ( )

−

−

1

314

14

rxyz

z

r

y

x

2222 1

1

=++, =













,=













−

−

θφcos

tan iif

if

x

y

x

x













+<



















−

0

0

1

tan π

• 
  

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Figure 10.3