Advanced Calculus fi..

326 Advanced Calculus, Fifth EditionBTHEOREM Inents in a domain D of space. The line integralLet u = Xi + Yj + Zk be a vector field with continuous compo-is independent of path in D if and only if there is a function F(x, y, z), defined inD such thatthroughout D. In other words the line integral is independent of path if and only ifu is a gradient vector:u = grad F.The proof for two dimensions in Section 5.6 can be repeated without essentialchange. When the integral is independent of path, X dx + Y dy + Z dz = dF forsome F, andas in the plane. 4THEOREM I1Let X, Y, Z be continuous in a domain D of space. The line integral,Jxdx+ydy+zdzis independent of path in D if and only ifon every simple closed curve C in D.This is proved as in the plane.THEOREM I11let X, Y, and Z have continuous partial derivatives in D. IfLet u = Xi + Yj + Zk be a vector field in a domain D of space;is independent of path in D, then curl u = 0 in D; that is,