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

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Unformatted text preview: Lecture 14 Spatial Laplacian in Spherical Coordinates Definition 1. The Laplacian is defined as = 2 x 2 + 2 y 2 + 2 z 2 The Laplacian is seen in equations such as Laplaces Equation: f = 0 (such f are called harmonic) Heat Equation 1 : f = f t Wave Equation 1 : f = 2 f t 2 Definition 2. The spherical polar coordinates are given by the equations x = r sin cos y = r sin sin z = r cos In terms of the spherical polar coordinates, the Laplacian is given by f = 2 f r 2 + 2 r f r + 1 r 2 2 f 2 + cot r 2 f + 1 r 2 sin 2 2 f 2 = 1 r 2 parenleftbigg r 2 f r parenrightbigg + 1 r 2 sin parenleftbigg sin f parenrightbigg + 1 r 2 sin 2 2 f 2 1 With appropriate units. 1 We shall seek solutions to f = 0 of the form f ( r, , ) = R ( r )( )( ) by separation of variables. We thus have the equation, 0 = f = r parenleftbigg r 2 R r parenrightbigg + 1 sin parenleftbigg sin R parenrightbigg + 1 sin 2 R Divide through by R to get 0 = 1 R r parenleftbigg r 2 R r parenrightbigg bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright PartA + 1 sin 1 parenleftbigg sin parenrightbigg + 1 sin 2...
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This note was uploaded on 07/08/2011 for the course MAT 6932 taught by Professor Staff during the Summer '10 term at University of Florida.

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

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