Advanced Calculus fi..

328 Advanced Calculus, Fifth Edition fA vector field u (whose components have continuous derivatives) such thatcurl u = 0in a domain D, is called irrotational in D. By virtue of the above theorems, irrotationalityin a simply connected domain is equivalent to each of the properties:IcUT ds = 0 for every simple closed curve in D;UT ds is independent of the path in D; .TS3u = grad F in D..I:,4 3ii;A theory similar to the preceding holds for surface integrals. Rather than give afull discussion here, we confine our attention to the counterpart of the last statementin Theorem 111; for more details we refer the reader to the books of Brand and Kellogglisted at the end of the chapter.ITHEOREM IVcontinuous partial derivatives in a spherical domain D. If div u = 0 in D:.; ALet u = Li + Mj + Nk be a vector field whose components have3then a vector field v = Xi + Yj + Zk in D can be found such thatthat is,curl v = u in D;Remark. The theorem provides a converse to the theorem of Section 3.6:div curl u = 0,for it asserts that if div u = 0, then u = curl v for some v. However, while div umay be 0 in an arbitrary domain Dl, the theorem provides v whose curl is u onlyin each spherical domain D contained in Dl, that is, no one v serves for all of Dl.Actually, the proof to follow gives one v for all of Dl when Dl is the interior ofa cube or of an ellipsoid or of any "convex" surface. The existence of v for all ofDl for more general cases can be established (see, for example, pages 203 ff. in the'book by Lamb listed at the end of the chapter).Vector fields u satisfying the condition div u = 0 are termed solenoidal. Thetheorem amounts to the assertion that solenoidal fields are (in suitable domains)fields of the form curl v, provided that the components of u have continuous partialderivatives. The field v is not unique (Problem 5).