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# gradient_fields - MIT OpenCourseWare http/ocw.mit.edu 18.02...

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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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V12. Gradient Fields in Space 1. The criterion for gradient fields. The curl in space. We seek now to generalize to space our earlier criterion (Section V2) for gradient fields in the plane. Criterion for a Gradient Field. Let F = M i + N j + P k be continuously differentiable. Then Proof. Since F = V f , when written out this says d f d f M = - , N = - . p = - . df therefore dx dy dz ' d M d2f - - - d2f - d N dy dydx dxdy dx ' The two mixed partial derivatives are equal since they are continuous, by the hypothesis that F is continuously differentiable. The other two equalities in (1) are proved similarly. Though the criterion looks more complicated to remember and to check than the one in two dimensions, which involves just a single equation, it is not difficult to learn and apply. For theoretical purposes, it can be expressed more elegantly by using the three-dimensional vector curl F. Definition. Let F = M i + N j + P k be differentiable. We define curl F by (3) curl F = (P, - N,) i + (M, - P,) j + (N, - My)k i j k = a, a, a, d (symbolic notation; d, = - etc.) dx' M N P d d d = V x F, where V = -i - k dx + -j + dy dz The equation (3) is the definition. The other two lines give symbolic ways of writing and of remembering the right side of (3). Neither the first nor second row of the determinant contains the sort of thing you are allowed to put into a determinant; however, if you "eval- uate" it using the Laplace expansion by the first row, what you get is the right side of (3).
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gradient_fields - MIT OpenCourseWare http/ocw.mit.edu 18.02...

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