The Chemistry Maths Book, Second Edition

9.4 Stationary points 253

EXAMPLES 9.3More partial derivatives

(i) The nonzero partial derivatives ofu 1 = 1 x 1 + 1 y

2

1 + 12 y

3

are

(ii) The first and second partial derivatives ofu 1 = 1 sin 1 x 1 cos 1 y 1 + 1 x 2 yare

andu

xy

1 = 1 u

yx

(iii) For the ideal gas,V 1 = 1 nRT 2 pand

0 Exercises 8–20

9.4 Stationary points

We saw in Section 4.10 that a functionf(x)of one variable has a stationary value

at pointx 1 = 1 aif its derivative at that point is zero; that is, iff′(a) 1 = 10. Geometrically,

the graph of the function has zero slope at the stationary point; its tangent line is

‘horizontal’. The corresponding condition for a function of two variables is that the

tangent planebe horizontal. A functionf(x, y)then has stationary point at(x, y) 1 = 1 (a, b)

if its partial first derivatives are zero:

(9.9)

orf

x

(a, b) 1 = 1 f

y

(a, b) 1 = 10. In view of equation (9.5), these are sufficient for all the first

derivatives of a continuous function to be zero at a point.

∂

∂

=

∂

∂

=,

f

x

f

y

0at ( )ab

Tn pn pT

V

p

nRT

p

V

T

nR

p

,,,

∂

∂













=− ,

∂

∂













=,

∂

2

VV

n

RT

∂ p













=

u

u

xy

xy

y

u

u

yx

x

xy yx

=

∂

∂∂

=− − =

∂

∂∂

=−

2

2

2

1

cos sin , cos ssiny

y

−

1

2

u

u

x

xy u

u

y

xy

xx yy

=

∂

∂

=− , =

∂

∂

=− +

2

2

2

2

2

sin cos sin cos

xx

y

3

u

u

x

xy

y

u

u

y

xy

x

y

xy

=

∂

∂

=+,=

∂

∂

cos cos =−sin sin −

1

2

∂

∂

=,

∂

∂

=+ ,

∂

∂

=+ ,

∂

∂

=

u

x

u

y

yy

u

y

y

u

y

1 2 6 2 12 12

2

2

2

3

3