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lecture26-09 - 18.02 Multivariable Calculus(Spring 2008...

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18.02 Multivariable Calculus (Spring 2008): Lecture 26 Green’s Theorem in normal forms. Green’s Theorem: why it works. Examples April 10 Reading Material: From Simmons : 21.3. Lecture Notes : V2, V3, V4. Last time: Green’s theorem. Flux. Today: Green’s Theorem in normal forms. Green’s Theorem: why it works. Examples 2 Elements from Lecture 25 We start by recalling the Green’s Theorem: Theorem 1. Assume that F = M ˆ i + N ˆ j is a continuous differentiable vector field and C is a closed simple curve around a region R . Then Work of F along C = C F · dr = C Mdx + Ndy = R N x - M y dA. Notation: line integral for work: C F · dr line integral for flux: C F · n ds = C G · dr, where G is obtained by rotating F by 90 . So in particular if F = M ˆ i + N ˆ j , then G = - N ˆ i + M ˆ j and C F · n ds = C G · dr = C - N dx + M dy. 3 Green’s Theorem for flux also called Green’s Theorem in normal form Since we defined the flux through the work we can now use the Green’s Theorem to relate the flux to a double integral. We have 1
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Theorem 2. With the same assumption of Green’s Theorem we have Flux through closed C = - N dx + M dy = R ∂M ∂x + ∂N ∂y dA.
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