The Chemistry Maths Book, Second Edition

9.8 Line integrals 275

9.8 Line integrals

Consider the functionF(x)and the integral

(9.47)

The integral was interpreted in Section 5.4 as the ‘area under the curve’ of the graph of

F(x)betweenx 1 = 1 aandx 1 = 1 b. An alternative interpretation is obtained by considering

the functionF(x)as some property associated with the points xon a line. For example,

letF(x)be the mass density of a straight rod of matter of lengthb 1 − 1 a(see Section 5.6).

The differential mass in element dxis then dm 1 = 1 F(x)dxand the total mass is

. Similarly (see Section 5.7), ifF(x)is the force acting on a body at

point xon a line, the differential work isdW 1 = 1 F(x)dxand the total work from ato b

is again the definite integral (9.47).

In these examples, a function is defined on a straight line, chosen to be the x-axis.

More generally, lety 1 = 1 f(x)represent a curve C in the xy-plane in the intervala 1 ≤ 1 x 1 ≤ 1 b,

as shown in Figure 9.7, and let the functionF(x, y)be some property associated with

the points on the curve. The quantity

is called a curvilinearor line integral, and the curve C is called the path of integration.

7

IfG(x, y)is a second function defined on the curve, the general line integral in the

plane is

IFxydxGxydy=,+, (9.48)













Z

C

() ()

Z

C

Fxydx(),

MFxdx

a

b

=Z ()

Z

a

b

Fxdx()

..

...

..

...

..

...

..

...

..

...

..

...

..

...

..

...

..

...

..

...

...

..

..

...

..

...

...

..

..

...

...

..

...

..

..

...

...

..

...

..

...

..

...

..

...

..

...

..

...

..

...

..

...

..

...

..

...

..

...

..

...

..

...

..

...

..

...

..

...

..

...

..

...

..

...

..

...

..

...

..

...

..

...

..

...

..

...

...

..

...

...

..

.

...

..

...

...

..

...

...

..

..........................................................................................................................................................................................................................................................................................

........

........

.

...

........

.......

....

y

x

• 
  


• 
  

b

a

a b

c

.

...

..

..

.

...

..

..

.

...

..

..

.

...

..

..

.

...

..

..

.

...

..

..

.

...

...

.

.

...

...

.

..

..

...

.

..

..

...

.

..

..

...

.

..

..

...

.

..

..

...

.

..

...

..

.

..

...

..

.

....................

..................

.....

......

.....

......

......

..

.......

..............

.............

..............

............

............

.............

............

...........

............

..........

...........

...........

..........

.........

...........

.........

.........

..........

.........

.........

.........

.........

........

.........

........

........

.........

........

.......

.........

.......

........

........

.......

.......

........

.......

.......

........

.......

.......

........

......

.......

.......

.......

.......

.......

......

.......

.......

......

.......

.......

..

Figure 9.7

7

The concept and notation of the line integral was used by Maxwell in 1855 in his studies of electric fields.

James Clerk Maxwell (1831–1879), born in Edinburgh, ranks with Newton and Einstein in pre-quantal theoretical

physics. Building on the work of Michael Faraday, Maxwell presented his field equations in his Dynamical theory

of the electromagnetic fieldin 1864. The Dynamical theory of gasesof 1859 describes the Maxwell distribution, with

applications to the theory of viscosity, the conduction of heat, and the diffusion of gases.

The line integral notation appeared in a physics text by Charles Delaunay (1816–1872) in 1856 for the work

done along a curve, and quickly became standard in physics.