Mathematical Methods for Physics and Engineering - Matematica.NET

11.4 CONSERVATIVE FIELDS AND POTENTIALS◮Evaluate the line integral∮I = [(e x y +cosx sin y) dx +(e x +sinx cos y) dy] ,around the ellipse x 2 /a 2 + y 2 /b 2 =1.CClearly, it is not straightforward to calculate this line integral directly. However, if we letP = e x y +cosx sin y and Q = e x +sinx cos y,then ∂P/∂y = e x +cosx cos y = ∂Q/∂x, andsoPdx+ Qdy is an exact differential (itis actually the differential of the function f(x, y) =e x y +sinx sin y). From the abovediscussion, we can conclude immediately that I =0.◭11.4 Conservative fields and potentialsSo far we have made the point that, in general, the value of a line integralbetween two points A and B depends on the path C taken from A to B. Intheprevious section, however, we saw that, for paths in the xy-plane, line integralswhose integrands have certain properties are independent of the path taken. Wenow extend that discussion to the full three-dimensional case.For line integrals of the form ∫ Ca · dr, there exists a class of vector fields forwhich the line integral between two points is independent of the path taken. Suchvector fields are called conservative. A vector field a that has continuous partialderivatives in a simply connected region R is conservative if, and only if, any ofthe following is true.(i) The integral ∫ BAa · dr, whereA and B lie in the region R, is independent ofthe path from A to B. Hence the integral ∮ Ca · dr around any closed loopin R is zero.(ii) There exists a single-valued function φ of position such that a = ∇φ.(iii) ∇×a = 0.(iv) a · dr is an exact differential.The validity or otherwise of any of these statements implies the same for theother three, as we will now show.First, let us assume that (i) above is true. If the line integral from A to Bis independent of the path taken between the points then its value must be afunction only of the positions of A and B. We may therefore write∫ BAa · dr = φ(B) − φ(A), (11.6)which defines a single-valued scalar function of position φ. If the points A and Bare separated by an infinitesimal displacement dr then (11.6) becomesa · dr = dφ,387