Flux integral of rectangle12/15/2023 ∫ C v ⋅ N ds = ∬ D ( P x + Q y ) d A = ∬ D 8 d A = 8 ( area of D ) = 80. Therefore, by the same logic as in Example 6.40, Let D be any region with a boundary that is a simple closed curve C oriented counterclockwise. The logic of the previous example can be extended to derive a formula for the area of any region D. In Example 6.40, we used vector field F ( x, y ) = 〈 P, Q 〉 = 〈 − y 2, x 2 〉 F ( x, y ) = 〈 P, Q 〉 = 〈 − y 2, x 2 〉 to find the area of any ellipse. Therefore, the area of the ellipse is π a b. d r = 1 2 ∫ C − y d x + x d y = 1 2 ∫ 0 2 π − b sin t ( − a sin t ) + a ( cos t ) b cos t d t = 1 2 ∫ 0 2 π a b cos 2 t + a b sin 2 t d t = 1 2 ∫ 0 2 π a b d t = π a b.r 4 ( t ) d t = ∫ a b P ( t, c ) d t + ∫ c d Q ( b, t ) d t − ∫ a b P ( t, d ) d t − ∫ c d Q ( a, t ) d t = ∫ a b ( P ( t, c ) − P ( t, d ) ) d t + ∫ c d ( Q ( b, t ) − Q ( a, t ) ) d t = − ∫ a b ( P ( t, d ) − P ( t, c ) ) d t + ∫ c d ( Q ( b, t ) − Q ( a, t ) ) d t.Therefore, the circulation of a vector field along a simple closed curve can be transformed into a double integral and vice versa. This form of the theorem relates the vector line integral over a simple, closed plane curve C to a double integral over the region enclosed by C. The first form of Green’s theorem that we examine is the circulation form. In particular, Green’s theorem connects a double integral over region D to a line integral around the boundary of D. Green’s theorem also says we can calculate a line integral over a simple closed curve C based solely on information about the region that C encloses. Green’s theorem says that we can calculate a double integral over region D based solely on information about the boundary of D. Green’s theorem takes this idea and extends it to calculating double integrals. Figure 6.32 The Fundamental Theorem of Calculus says that the integral over line segment depends only on the values of the antiderivative at the endpoints of.
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