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\fDef. A vector field F is said to be conservative if it is the gradient of some scalar field, i.e., if there exists a scalar field f such that F = Vf . Such a scalar field f is called a potential field of F . Thm. 1 If F is conservative, then F is irrotational. Thm. 2 Let F(t), t : a -> b be a parametrization of a piecewise smooth curve C . If f is continuously differentiable in some neighborhood of C, then [VS .Ids = f(F(b))-f (F(a)). Thm. 3 The tangential integral of a conservative vector field is path-independent. Note: The tangential integral of a vector field is path-independent if and only if the tangential integral along all closed paths is zero. Thus we have the following Corollary If F is conservative, then & F . 7 ds =0 for every piecewise smooth closed curve C on the domain of F . Def. A region is said to be path-wise connected if every pair of points on the region can be connected by a piecewise smooth curve that lies entirely on the region. Thm. 4 Suppose that the domain of F is an open and path-wise connected region. If F is continuous and the tangential integral of F is path-independent, then F is conservative. Proof We will use the constructive-proof technique to establish a potential field f of F as follows. Thm. 5 Let F be a continuously differentiable vector field defined on a simply-connected region. If F is irrotational, then F is conservative. Proof Let C be an arbitrary piecewise smooth simple closed curve in the domain of F . We will demonstrate that & F. 7 ds =0. This will imply that F is conservative according to the corollary to Thm.4. Since the domain of F is simply-connected, C is the boundary of an orientable surface S that lies entirely in the domain of F . Since S lies entirely in the domain of F , F is continuously differentiable on S . Thus Stoke's Theorem can be applied to S & C, and yields { F. f ds = [[ (VxF).Ads Since F is irrotational, V xF = 0. Hence & F.fds = [[ o.ads =0