lecture23

lecture23 - MATH100 Def: Lecture 23 Divergence Theorem If F...

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2006 Fall MATH 100 Lecture 23 1 MATH100 Lecture 23 Divergence Theorem ( ) ( ) ( ) ( ) Def: If , , , , , , , , , then the divergence is defined by div F xyz f xyzi g xyz j hxyzk fg h F xyz =++ ∂∂∂ G G G G G ( ) 2 3 3 2 2 2 3 4 2 2 2 4 div 3 : Example y x y z y F k z y x j y xz i z xy F + + = + + + = G G G G G
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2006 Fall MATH 100 Lecture 23 2 MATH100 Lecture 23 Divergence Theorem () Divergence Theorem Let be a solid surface oriented by outward unit normals. I f ,, and , and exist, then div xy z G G F xyz f xyzi g xyz j hxyzk fg h F ndS FdV σ =++ ⋅= ∫∫ ∫∫∫ G G GG G
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2006 Fall MATH 100 Lecture 23 3 MATH100 Lecture 23 Divergence Theorem Proof: To prove with simple solid only (continue next page)
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2006 Fall MATH 100 Lecture 23 4 MATH100 Lecture 23 Divergence Theorem () ( ) ( ) { } 12 , , : , , for , we try to prove , , , , , , Withou V V V Gx y z g x y z gx y x yR f f x y z i ndS dV x g g x y z j ndS dV y h h x y z k ndS dV z σ ≤≤ ⋅= ∫∫ ∫∫∫ G G G G G G t loss of generality, we prove the last equality. (continue next page)
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2006 Fall MATH 100 Lecture 23 5 MATH100 Lecture 23 Divergence Theorem () () () ( ) () 123 3 2 2 2 We have , , , , , note that 0 while , , , , , , , , R R h x y z k ndS h x y z k ndS zz h x y z k ndS h x y g x y k i j k dA xy hxyg xy dA σσ σ ⋅= + + = ⎛⎞ ∂∂ −− + ⎜⎟ ⎝⎠ = ∫∫ ∫∫ ∫∫ ∫∫ GG G G ) 1 1 1 , , , , , , , , R R h x y z k ndS h x y g x y k i j k dA + =− G G
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2006 Fall MATH 100 Lecture 23 6 MATH100 Lecture 23 Divergence Theorem ( ) ( ) ( ) ( ) ( ) [ ] () [] ∫∫∫ ∫∫ ∫ ∫∫ ∫∫ ∫∫ =
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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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lecture23 - MATH100 Def: Lecture 23 Divergence Theorem If F...

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