Vector_Notes_Week_2 - EM Research Group Cartesian...

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EM Research Group Cartesian (Rectangular) to Cylindrical Transformation Coordinate Transformation z z x y tan y x 1 2 2 = = φ + = ρ x y z ρ ˆ φ ˆ z ˆ ( ) ( ) r r P z , , P = φ ρ r r φ ρ z Unit Vector Transformation () () z ˆ z ˆ cos y ˆ sin x ˆ ˆ sin y ˆ cos x ˆ ˆ = φ + φ = φ φ + φ = ρ
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EM Research Group Cartesian/Cylindrical Conversion (continued) () ( ) ( ) ( ) ( ) ) () () ( ) 1 z ˆ z ˆ 0 ˆ z ˆ 0 ˆ z ˆ 0 z ˆ y ˆ cos ˆ y ˆ sin ˆ y ˆ 0 z ˆ x ˆ sin ˆ x ˆ cos ˆ x ˆ = = φ = ρ = φ = φ φ = ρ = φ = φ φ = ρ z y x A z ˆ A y ˆ A x ˆ + + = A r ( ) ( ) ( ) ( ) ( ) () () () () () z z y x z y x z y x y x z y x A A z ˆ z ˆ A y ˆ z ˆ A x ˆ z ˆ z ˆ A cos A sin A A z ˆ ˆ A y ˆ ˆ A x ˆ ˆ ˆ A sin A cos A A z ˆ ˆ A y ˆ ˆ A x ˆ ˆ ˆ A = + + = = φ + φ = φ + φ + φ = φ = φ + φ = ρ + ρ + ρ = ρ = φ ρ A A A r r r z A z ˆ A ˆ A ˆ + φ + ρ = φ ρ A r ( ) ( ) ( ) ( ) ( ) () () () ( ) ( ) z z z z y z x A A z ˆ z ˆ A ˆ z ˆ A ˆ z ˆ z ˆ A cos A sin A A z ˆ y ˆ A ˆ y ˆ A ˆ y ˆ y ˆ A sin A cos A A z ˆ x ˆ A ˆ x ˆ A ˆ x ˆ x ˆ A = + φ + ρ = = φ + φ = + φ + ρ = = φ φ = + φ + ρ = = φ ρ φ ρ φ ρ φ ρ φ ρ A A A r r r Cartesian-to-Cylindrical Cylindrical-to-Cartesian … some facts…
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EM Research Group Example: Position Vector Goal: Convert position vector from Cartesian to Cylindrical Coordinates () ( ) z z ˆ cos y sin x ˆ sin y cos x ˆ r z ˆ cos r sin r ˆ sin r cos r ˆ z z ˆ y y ˆ x x ˆ z y x y x + φ + φ φ + φ + φ ρ = + φ + φ φ + φ + φ ρ = + + = r r r r r r Recall… φ ρ = φ ρ = sin y cos x Hence… ( ) z z ˆ ˆ z z ˆ cos sin cos sin ˆ sin cos ˆ 2 2 + ρ ρ = + φ φ ρ + φ φ ρ φ + φ ρ + φ ρ ρ = r r r r
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EM Research Group Conversion of Cylindrical to Rectangular Components and Coordinates Convert the following vector field from cylindrical to Cartesian components and coordinates : facts… () 2 2 2 y x y sin x cos + = ρ ρ = φ ρ = φ () ( ) ( ) 5 z ˆ 2 ˆ cos 3 ˆ ρ φ φ ρ = r A r r ( ) ( ) ()( ) ( ) φ ρ + φ = + φ ρ φ φ = + ρ φ φ ρ = = sin 2 cos 3 0 5 sin 2 cos cos 3 5 z ˆ x ˆ 2 ˆ x ˆ cos 3 ˆ x ˆ x ˆ A 2 x A r ( ) ( ) ()() ( ) () () () φ ρ φ φ = + φ ρ φ φ = + ρ φ φ ρ = = cos 2 cos sin 3 0 5 cos 2 sin cos 3 5 z ˆ y ˆ 2 ˆ y ˆ cos 3 ˆ y ˆ y ˆ A y A r ( ) ( ) 5 1 5 0 2 0 cos 3 5 z ˆ z ˆ 2 ˆ z ˆ cos 3 ˆ z ˆ z ˆ
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This note was uploaded on 09/27/2009 for the course ECE 280 taught by Professor Mukkamala/udpa during the Spring '08 term at Michigan State University.

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Vector_Notes_Week_2 - EM Research Group Cartesian...

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