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USEFULMATHEMATICALRELATIONS

USEFULMATHEMATICALRELATIONS - USEFUL MATHEMATICAL RELATIONS...

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USEFUL MATHEMATICAL RELATIONS A) Coordinate Transformations: Rectangular–to–Spherical (and Vice–Versa): ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) sin cos cos cos sin sin sin cos sin cos cos sin x R y R z R a a a a a a a a a a a θ ϕ θ ϕ θ θ ϕ θ ϕ ϕ θ ϕ θ ϕ ϕ θ θ = + = + + = K K K K K K K K K K K ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) sin cos sin sin cos cos cos cos sin sin sin cos R x y z x y z x y a a a a a a a a a a a θ ϕ θ ϕ θ ϕ θ θ ϕ θ ϕ θ ϕ ϕ = + + = + = − + K K K K K K K K K K K Rectangular–to–Cylindrical (and Vice–Versa): ( ) ( ) ( ) ( ) cos sin sin cos x r y r z z a a a a a a a a ϕ ϕ ϕ ϕ ϕ ϕ = = + = K K K K K K K K ( ) ( ) ( ) ( ) cos sin sin cos r x y x y z z a a a a a a a a ϕ ϕ ϕ ϕ ϕ = + = − + = K K K K K K K K Cylindrical–to–Spherical (and Vice–Versa): ( ) ( ) ( ) ( ) sin cos cos sin R r z r z a a a a a a a a θ ϕ ϕ θ θ θ θ = + = = K K K K K K K K ( ) ( ) ( ) ( ) sin cos cos sin r R z R a a a a a a a a θ ϕ ϕ θ θ θ θ θ = + = = K K K K K K K K B) Trigonometric Relations: ( ) ( ) ( ) ( ) ( ) sin sin cos cos sin α β α β α β = ( ) ( ) ( ) ( ) ( ) cos cos cos sin sin α β α β α β ± =
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( ) sin cos 2 π α α =
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