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# 14a - Lecture 14 Spatial Laplacian in Spherical Coordinates...

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Lecture 14 Spatial Laplacian in Spherical Coordinates Definition 1. The Laplacian Δ is defined as Δ = 2 ∂x 2 + 2 ∂y 2 + 2 ∂z 2 The Laplacian is seen in equations such as Laplace’s Equation: Δ f = 0 (such f are called harmonic) Heat Equation 1 : Δ f = ∂f δt Wave Equation 1 : Δ f = 2 f ∂t 2 Definition 2. The spherical polar coordinates are given by the equations x = r sin θ cos φ y = r sin θ sin φ z = r cos θ In terms of the spherical polar coordinates, the Laplacian is given by Δ f = 2 f ∂r 2 + 2 r ∂f ∂r + 1 r 2 2 f ∂θ 2 + cot θ r 2 ∂f ∂θ + 1 r 2 sin 2 θ 2 f ∂φ 2 = 1 r 2 parenleftbigg r 2 ∂f ∂r parenrightbigg + 1 r 2 sin θ ∂θ parenleftbigg sin θ ∂f ∂θ parenrightbigg + 1 r 2 sin 2 θ 2 f ∂φ 2 1 With appropriate units. 1

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We shall seek solutions to Δ f = 0 of the form f ( r, θ, φ ) = R ( r )Θ( θ )Φ( φ ) by ’separation of variables’. We thus have the equation, 0 = Δ f = ∂r parenleftbigg r 2 ∂R ∂r ΘΦ parenrightbigg + 1 sin θ ∂θ parenleftbigg sin θR Θ ∂θ Φ parenrightbigg + 1 sin 2 θ R ΘΦ ′′ Divide through by R ΘΦ to get 0 = 1 R ∂r parenleftbigg r 2 ∂R ∂r parenrightbigg bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright P artA + 1 sin θ 1 Θ ∂θ parenleftbigg sin θ Θ ∂θ parenrightbigg + 1 sin 2 θ Φ ′′ Φ bracehtipupleft
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14a - Lecture 14 Spatial Laplacian in Spherical Coordinates...

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