Mathematical Methods for Physics and Engineering : A Comprehensive Guide

11.4 CONSERVATIVE FIELDS AND POTENTIALS

Evaluate the line integral

I=

∮

C

[(exy+cosxsiny)dx+(ex+sinxcosy)dy],

around the ellipsex^2 /a^2 +y^2 /b^2 =1.

Clearly, it is not straightforward to calculate this line integral directly. However, if we let

P=exy+cosxsiny and Q=ex+sinxcosy,

then∂P /∂y=ex+cosxcosy=∂Q/∂x,andsoPdx+Qdyis an exact differential (it
is actually the differential of the functionf(x, y)=exy+sinxsiny). From the above
discussion, we can conclude immediately thatI=0.

11.4 Conservative fields and potentials

So far we have made the point that, in general, the value of a line integral

between two pointsAandBdepends on the pathCtaken fromAtoB.Inthe

previous section, however, we saw that, for paths in thexy-plane, line integrals

whose integrands have certain properties are independent of the path taken. We

now extend that discussion to the full three-dimensional case.

For line integrals of the form

∫
Ca·dr, there exists a class of vector fields for
which the line integral between two points isindependentof the path taken. Such

vector fields are calledconservative. A vector fieldathat has continuous partial

derivatives in a simply connected regionRis conservative if, and only if, any of

the following is true.

(i) The integral

∫B

Aa·dr,whereAandBlie in the regionR, is independent of

the path fromAtoB. Hence the integral

∮

Ca·draround any closed loop

inRis zero.

(ii) There exists a single-valued functionφof position such thata=∇φ.

(iii)∇×a= 0.

(iv)a·dris an exact differential.

The validity or otherwise of any of these statements implies the same for the

other three, as we will now show.

First, let us assume that (i) above is true. If the line integral fromAtoB

is independent of the path taken between the points then its value must be a

function only of the positions ofAandB. We may therefore write

∫B

A

a·dr=φ(B)−φ(A), (11.6)

which defines a single-valued scalar function of positionφ. If the pointsAandB

are separated by an infinitesimal displacementdrthen (11.6) becomes

a·dr=dφ,