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Unformatted text preview: 18.02 Multivariable Calculus (Spring 2009): Lecture 34 Stokes Theorem May 1 Reading Material: From Simmons : 21.5. From Lecture Notes : V13. Last time: Line Integrals, conservative vector fields, Curl and potential functions in 3D Today: Stokes Theorem. 2 The theorems we know We recall here the Greens Theorem in Normal Form for flux, its generalization in 3D, that is the Divergence Theorem, and the Greens Theorem for work in 2D. Greens Theorem in Normal Form : This is a theorem in 2D and it is used to compute the flux of a vector field ~ F = M i + N j through a simple, piecewise smooth and closed curve in terms of a double integral over the enclosed region R , more precisely Theorem 1. Given a continuously differentiable vector field ~ F = M i + N j and a clockwise oriented simple, piecewise smooth and closed curve C enclosing a region R Flux across C = Z C ~ F nds = Z C ( N i- M j ) d~ r = ZZ R [ M x + N y ] dA. (2.1) As we know there is a generalization of this in 3D Divergence Theorem : This theorem is used to compute the flux of a 3D vector filed ~ F = M i + N j + P k across an oriented closed surface S by a triple integral over the space region D enclosed by S . More precisely 1 Theorem 2. Assume the 3D vector filed...
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