Mathematical Methods for Physics and Engineering - Matematica.NET

11.1 LINE INTEGRALS◮Evaluate the line integral I = ∮ xdy,whereC is the circle in the xy-plane defined byCx 2 + y 2 = a 2 , z =0.Adopting the usual convention mentioned above, the circle C is to be traversed in theanticlockwise direction. Taking the circle as a whole means x is not a single-valuedfunction of y. We must therefore divide the path into two parts with x =+ √ a 2 − y 2 forthe semicircle lying to the right of x =0,andx = − √ a 2 − y 2 for the semicircle lying tothe left of x = 0. The required line integral is then the sum of the integrals along the twosemicircles. Substituting for x, itisgivenby∮ ∫ a √∫ −aI = xdy = a2 − y 2 dy +(− √ )a 2 − y 2 dyC−aa∫ a √=4 a2 − y 2 dy = πa 2 .0Alternatively, we can represent the entire circle parametrically, in terms of the azimuthalangle φ, sothatx = a cos φ and y = a sin φ with φ running from 0 to 2π. The integral cantherefore be evaluated over the whole circle at once. Noting that dy = a cos φdφ,wecanrewrite the line integral completely in terms of the parameter φ and obtain∮ ∫ 2πI = xdy = a 2 cos 2 φdφ= πa 2 . ◭C011.1.2 Physical examples of line integralsThere are many physical examples of line integrals, but perhaps the most commonis the expression for the total work done by a force F when it moves its pointof application from a point A to a point B along a given curve C. We allow themagnitude and direction of F to vary along the curve. Let the force act at a pointr and consider a small displacement dr along the curve; then the small amountof work done is dW = F · dr, as discussed in subsection 7.6.1 (note that dW canbe either positive or negative). Therefore, the total work done in traversing thepath C is∫W C = F · dr.CNaturally, other physical quantities can be expressed in such a way. For example,the electrostatic potential energy gained by moving a charge q along a path C inan electric field E is −q ∫ CE · dr. We may also note that Ampère’s law concerningthe magnetic field B associated with a current-carrying wire can be written as∮B · dr = µ 0 I,Cwhere I is the current enclosed by a closed path C traversed in a right-handedsense with respect to the current direction.Magnetostatics also provides a physical example of the third type of line381