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MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable Calculus Fall 200 7 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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V2. Gradient Fields and Exact Differentials 1. Criterion for gradient fields. Let F = M(x, y) i + N(x, y) j be a two-dimensional vector field, where M and N are continuous functions. There are three equivalent ways of saying that F is conservative, i.e., a gradient field: (1) F = V f H R F . dr is path-independent H F . dr = 0 for any closed C Unfortunately, these equivalent formulations don't give us any effective way of deciding if a given field F is a conservative field or not. However, if we assume that F is not just continuous but is even continuously differentiable (meaning: M, , My,N, , Ny all exist and are continuous), then there is a simple and elegant criterion for deciding whether or not F is a gradient field in some region. Criterion. Let F = M i + N j be continuously differentiable in a region D. Then, in D, (2) F = V f for some f (x, y) My = N, . Proof. Since F = Vf , this means M = f, and N = fy . Therefore, M y = f x y and N,=fy,. But since these two mixed partial derivatives are continuous (since My and N, are, by hypothesis), a standard 18.02 theorem says they are equal. Thus My = N,. This theorem may be expressed in a slightly different form, if we define the scalar function called the two-dimensional curl of F by (3) curl F = N, - M, Then (2) becomes This criterion allows us to test F to see if it is a gradient field. Naturally, we would also like to know that the converse is true: if curl F = 0, then F is a gradient field. As we shall see, however, this requires some additional hypotheses on the region D. For now, we will assume D is the whole plane. Then we have Converse t o Criterion. Let F = M i + N j be continuously differentiable for all x, y. (4) My = N, for all x, y F = V f for some differentiable f and all x, y.
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