The Chemistry Maths Book, Second Edition

272 Chapter 9Functions of several variables

Therefore,

0 Exercises 50–52

EXAMPLE 9.20Show that the functionf 1 = 1 ln 1 r, where , satisfies the

Laplace equation.

Because the function depends on ronly,∂f 2 ∂θ 1 = 10 and the derivatives with respect

to rare total derivatives:

Nowdf 2 dr 1 = 112 randd

2

f 2 dr

2

1 = 1 − 12 r

2

. Therefore∇

2

f 1 = 10.

This example is important because it can be shown that the only solution of the

Laplace equation in two dimensions that depends on alone has the

general formf 1 = 1 a 1 ln 1 r 1 + 1 b. This function occurs in potential theory in two dimensions.

9.7 Exact differentials

One of the fundamental equations of thermodynamics, combining both the first and

second laws, is

dU 1 = 1 TdS 1 − 1 pdV (9.40)

where Uis the internal energy of a thermodynamic system, Sis its entropy, and p, V,

and Tare the pressure, volume, and temperature.*The quantity dUis the total

differential ofU 1 = 1 U(S, V)as a function of Sand V. It can therefore be written as

(9.41)

so that, equating (9.40) and (9.41),

(9.42)

The expression on the right side of (9.40) is called an exact differential. In general, a

differential

F(x, y)dx 1 + 1 G(x, y)dy (9.43)

T

U

S

p

U

V

VS

=

∂

∂













,−=

∂

∂













dU

U

S

dS

U

V

dV

VS

=

∂

∂













• 
  

∂

∂













rxy=+

22

∇=+

2

2

2

1

fr

df

dr

r

df

dr

()

rxy=+

22

∂

∂

• 
  

∂

∂

• 
  

∂

∂

=+−=

2

22

2

2222

1 1 224

0

f

r

r

f

r

r

ff

r

f

r

f

θ r

*For a single closed phase with constant composition. More generally, the equation applies to each separate

phase, with additional terms if the amounts of substance are not constant.