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# Course Summary - AMATH 231 COURSE SUMMARY Curves Vector...

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AMATH 231 COURSE SUMMARY Curves & Vector Fields [ C h . 1 ] - Curves in R n : n R b a g ] , [ : )) ( ),. .., ( ), ( ( ) ( 2 1 t g t g t g t g n = (ie path of a particle through space with parameter time t ) - Expressing Equations in Parametric Form (above) - Limits: L t g t t = ) ( lim 0 means that 0 || ) ( || 0 lim = L t g t t - Continuity: g is continuous at t 0 means that ) ( ) ( 0 0 t g t g t t lim = where ) ( 0 t g is defined a n d ) ( 0 t t t lim g exists. - Derivatives: t t g t t g t g t + = ) ( ) ( lim ) ( 0 - Arclength: = b a dt t g s || ) ( || - Scalar Field: R R f n : ) ( x f - Vector Field: m n R R F : )) ( ),. .., ( ), ( ( ) ( 2 1 x F x F x F x F m = - Sketching Field Lines: If given ) , ( y x F , use )) ( ( ) ( t g F t g = where )) ( ), ( ( ) ( t y t x t g = Line Integrals & Green's Theorem [ C h . 2 ] - Line Integral of a Scalar Field: If curve C in R n is defined by ] , [ ), ( b a t t g x = where g is C 1 and scalar field f is continuous on C , then the line integral of f along C is: = b a C dt t g t g f ds f || ) ( || )) ( ( (total amount of "stuff" along C) - Line Integral of a Vector Field: If curve C in R n is defined by ] , [ ), ( b a t t g x = where g is C 1 and vector field F is continuous on C , then the line integral of F along C is: = b a C dt t g t g F x d F ) ( )) ( ( (total amount of F x along C) - Properties: + = C C C x d G x d F x d G F ) + ( = C C x d F k x d F k ) ( + = 2 1 2 1 C C C C x d F x d F x d F = C C x d F x d F - Conservative (Gradient) Fields: - First Fundamental Theorem: Let U be connected and open. Let n R n R U F : be a continuous vector field with a path-independent line integral in U . I f x x x d F x 0 ) ( = φ for some point x 0 , then U x x F x = φ ) ( ) ( . - Second Fundamental Theorem: Let U be connected and open. Let n R n R U F : be a continuous vector field in U . I f φ = F , where is a C R C φ : 1 C is any curve joining x 1 x 2 in U , t h e n )

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- Simple Closed Curves & Path-Independence: Let U be connected and open. A continuous vector field n R n R U F : is path-independent in U if and only if 0 = C x d F for every simple closed curve in U . - Simply-Connected Sets:
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Course Summary - AMATH 231 COURSE SUMMARY Curves Vector...

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