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2_dimentnl_flux

2_dimentnl_flux - 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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V3. Two-dimensional Flux In this section and the next we give a different way of looking at Green's theorem which both shows its significance for flow fields and allows us to give an intuitive physical meaning for this rather mysterious equality between integrals. We have seen that if F is a force field and C a directed curve, then (1) work done by F along C = In words, we are integrating F . T, the tangential component of F, along the curve C. In component notation, if F = M i + N j , then the above reads Analogously now, we may integrate F . n, the normal component of F along C. To describe this, suppose the curve C is parametrized by the arclength s, increasing in the positive direction on C. The position vector for this parametrization and its corresponding tangent vector are given respectively by where we have used t instead of T since it is a unit vector- its length is 1, as one can see by dividing through by ds on both sides of ds = J(dx)~ + ( d ~ ) ~ .
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