notes16 2317 - Curl of a Vector ECE 2317 Applied...

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Prof. Filippo Capolino ECE Dept. Fall 2006 Notes 16 ECE 2317 ECE 2317 Applied Electricity and Magnetism Applied Electricity and Magnetism Notes prepared by the EM group, University of Houston . (used by Dr. Jackson, spring 2006) Curl of a Vector Curl of a Vector ± ± 0 0 0 1 lim 1 lim 1 lim x y z C s C s C s xcu r lV V d r S yc u r Vd r S zc u r r S Δ→ ⋅≡ Δ Δ Δ ± v v v ( ) , , vector function Vxyz = vector function curl V = x y z C x C y C z Δ S Δ S Δ S Note: paths are defined according to the “right-hand rule” Curl of a Vector (cont.) Curl of a Vector (cont.) “curl meter” ˆˆ ˆ ,, xy z = ± A velocityof rotation (in the sense indicated) curl V ⋅= ± A Assume that V represents the velocity of a fluid. Curl Calculation Curl Calculation y z Δ y Path C x : z 1 2 3 4 C x 0, , 0 2 0, , 0 2 0, 0, 2 0, 0, 2 xx xyzz CC z y y y V dr V dx V dy V dz V z y Vz z Vy z Δ ⎛⎞ + + ≈ Δ ⎜⎟ ⎝⎠ Δ −− Δ Δ +− Δ Δ −Δ ∫∫ vv (1) (2) (3) (4)
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() 0, ,0 22 0, 0, 0, 0, x x zz C yy y z y z C VV Vd r yz y zy z V V SS V V r S ⎡Δ Δ ⎛⎞ −− ⎜⎟ ⎢⎥ ⎝⎠ ⋅≈ Δ Δ Δ ⎣⎦ Δ −ΔΔ Δ −Δ ∂∂ Δ v v Curl Calculation (cont.) Curl Calculation (cont.) x y z C V V r S v Δ ± 0 1 lim x C s xcu r lV r S v Δ→ ⋅≡ Δ ± y z V V xc u r ⋅= From the curl definition: Hence Curl Calculation (cont.) Curl Calculation (cont.) Similarly, y z x z C y x C V V r S zx V V r S xy v v Δ Δ ± ± xx curl V x y z =− +− ± Hence, ± x z V V yc u r z x y x V V zc u r ± Curl Calculation (cont.) Curl Calculation (cont.) Del Operator Del Operator ± ± ± ± ± ± ± ± xyz x x Vx y z x V y V z V VVV xz ∂∂∂ ∇× = + + × + + = ±± ± ± ± ± ∇≡ + + ±
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Del Operator (cont.) Del Operator (cont.) curl V V =∇× Hence, Example Example ± () ± 22 3 32 2 Vx x y z yx z zx z =+ + ± ± ± 03 2 3 4 6 z yz x y x y z ∇× = + + ± ± ± yy x x zz VV y z xz xy ∂∂ ⎛⎞ × =− −− +− ⎜⎟ ⎝⎠ ± Example Example ± ( ) ± ± 1 x x y y z Vz = = + = ± ± x y Example (cont.) Example (cont.) 1 = ± 10 ± ⋅ =− < x y
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Summary of Curl Formulas Summary of Curl Formulas l ± () 11 zz V VV V Vz φ φρ ρ ρφ ⎛⎞ ∂∂ ⎜⎟ ∇× = + + ⎝⎠ ± ( ) ± ± sin 1 1 sin sin rr Vr V rV V r r r φφ θ θθ ⎡⎤ = + + ⎢⎥ ⎣⎦ ± ± ± yy x x Vx y z yz xz xy × =− −− +− ± Stokes Stokes ’ Theorem Theorem ± n
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notes16 2317 - Curl of a Vector ECE 2317 Applied...

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