lecture22

lecture22 - MATH 100 Lecture 22 Introduction to surface...

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2006 Fall MATH 100 Lecture 22 1 MATH 100 Lecture 22 Introduction to surface integrals () then , , , be density let the : lamina bent a of Mass z y x δ ( ) ( ) ∫∫ + + = = R y x dA f f y x f y x M M R xy y x f z z y x 1 , , , by defined is lamina the of mass the then , region the is plane on the lamina this of projection the if and ; , equation the has , , density with lamina curved a If : Def 2 2
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2006 Fall MATH 100 Lecture 22 2 Definition of density function: () ,, l i m where is the samll section of the area containing , , M xyz S Sx y z δ Δ = Δ Δ ∫∫ + + = R y x dA f f S 1 is area the and area, surface the to equals mass the , 1 when : Remark 2 2
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2006 Fall MATH 100 Lecture 22 3 () ( ) k k k y k k x k k k k k k k k k A y x f y x f y x f y x S z y x M Δ + + = Δ = Δ 1 , , , , , , , 2 2 δ MATH 100 Lecture 22 Introduction to surface integrals
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2006 Fall MATH 100 Lecture 22 4 MATH 100 Lecture 22 Introduction to surface integrals () 1 22 1 Thus , , , , , 1 , , , 1 n k k n kk x y k k xy R MM x yf x x x y A xyf xy f f dA δ = ∗∗ = ≈Δ ≈+ + Δ →+ + ∫∫
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2006 Fall MATH 100 Lecture 22 5 ( ) mass its find , , , , 1 0 , 1 Ex 0 2 2 δδ = + = z y x z y x z () = + = + = + = + + = = = + ∫∫ ∫∫ 1 5 6 1 4 4 1 3 2 1 4 2 1 2 1 4 1 2 2 2 , 2 , 1 : : Sol 2 3 0 1 0 2 3 0 1 0 0 2 0 1 0 2 0 2 2 0 2 2 πδ θ δ π u du u rdrd r dA y x M y f x f y x R R y x MATH 100 Lecture 22 Introduction to surface integrals
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2006 Fall MATH 100 Lecture 22 6 ( ) {} 1 1 Surface integral: Let be a surface with finite surface area and , , a continuous function defined on . Subdivide into patches with surface area and sum up n i i n i gxyz S σ σσ = = Δ () = Δ n k k k k k S z y x g 1 , , () () k n k k k k n S z y
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This note was uploaded on 09/17/2011 for the course MATH 100 taught by Professor Qt during the Fall '09 term at HKUST.

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lecture22 - MATH 100 Lecture 22 Introduction to surface...

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