grad_fld_n_diff

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MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable Calculus Fall 2007 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 = Mi + Nj 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, My=fxy 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 = - 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 to Criterion. Let F = + Nj 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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This note was uploaded on 05/04/2011 for the course MATH 18.02 taught by Professor Auroux during the Spring '08 term at MIT.

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grad_fld_n_diff - MIT OpenCourseWare http:/ocw.mit.edu...

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