Lecture 3

# Lecture 3 - ∂ ∂ × ∇ A z y x A A A right hand rule low velocity Flow velocity vorticity agnetic field Magnetic field current xyz xy x u u u A

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ivergence divergence ven a vector field the divergence given a vector field, the divergence operation tells if there is a source or sink useful for relating electric fields to charges vector field ==> scalar A A = div ds A lim v v Δ Δ 0
Cartesian Coordinates z A y A x A z y x + + = A

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source
ample example

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Gauss’s law or divergence theorem ds A lim v v Δ Δ 0 A = ds A Δ dv v A A onvert from volume integral to Δ v convert from volume integral to surface integral

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= ds A A dv Δ v x u A = x dv A ( ) ( ) 1 1 1 = = 0 1 x Δ v ds A 1 = [ ] 1 1 =
Curl place paddle wheel in a river o rotation at no rotation at the center rotation at the dges edges

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dl A u n lim s s Δ Δ 0 A A × = curl the vector u n is out of e screen the screen right hand rule ± Δ s is surface enclosed within loop s Δ closed line integral
artesian coordinates Cartesian coordinates z y x u u u z y x

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Unformatted text preview: ∂ ∂ × ∇ A z y x A A A right hand rule low velocity Flow velocity vorticity agnetic field Magnetic field current xyz xy x u u u A + + = z y x z y x u u u ∂ ∂ ∂ = × ∇ A z z y x ∂ ∂ ∂ xyz xy x x u A xz = × ∇ y yz u − z y u + • ∫ dl A u n lim s s Δ ⇒ Δ ≡ × ∇ A • ≈ Δ • × ∇ dl A u u A n n s ∫ • l s ∫ ∫ × ∇ dl A u u A n n ds ∫ ∫ • = • × ∇ dl A A ds ∫ ∫ • = • × ∇ dl A u u A n n ds xyz xy x u u u A + + = z y x y yz xz u u u A + − = × ∇ x u x z y x 1 1 ∫ • dl A ∫ = xdx ( ) ∫ = + 1 ydy x ∫ + 1 xdx ( ) ∫ = + 1 ydy x 2 1 2 1 2 1 − − + = 2 1 = ∫ • × ∇ dxdy A ∫ = dxdy y 2 1 =...
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## This note was uploaded on 02/04/2010 for the course EEGR 304 taught by Professor Craigscott during the Spring '08 term at Morgan.

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Lecture 3 - ∂ ∂ × ∇ A z y x A A A right hand rule low velocity Flow velocity vorticity agnetic field Magnetic field current xyz xy x u u u A

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