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Lecture 3 - ∂ ∂ × ∇ A z y x A A A right hand rule...

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divergence given 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
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Cartesian Coordinates A A A z y x z y x + + = A
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source
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example
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Gauss’s law or divergence theorem ds A lim A v v Δ Δ 0 = ds A Δ Δ dv v A A convert from volume integral to v convert from volume integral to surface integral
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= Δ ds A A dv v x x u A = dv A ( ) ( ) 1 1 1 = = 0 1 x Δ v [ ] 1 1 ds A 1 = =
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Curl place paddle wheel in a river no rotation at the center rotation at the edges
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dl A u n lim A A l s s Δ Δ 0 × = curl the vector u n is out of the screen right hand rule Δ s is surface enclosed within loop s Δ closed line integral
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Cartesian coordinates z y x u u u = × A A A A z y x z y x
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right hand rule Flow velocity ti it vorticity Magnetic field t current
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z y xyz xy x u u u A x + + = z y x u u u = × A xyz xy x z y x x u A xz = × y yz u z y u +
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dl A u A n lim s s Δ Δ
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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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