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MATH529: L11 BVPs in curved coordinate systems
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Polar coordinates
Cylindrical coordinates
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Polar coordinates
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, , ,
Lamé (metric) coefficients:
In Cartesian coordinates correspond to actual traversed distances
In Polar coordinates traversed distances are
Nabla operator:
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Differential operators in polar coordinates
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Gradient of scalar function
Divergence of vector function
Laplacian of scalar function
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Laplace equation in polar coordinates
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Dirichlet problem: find in a disk of radius
Separation of variables:
Solution is periodic w.r.t , iff , say
Along : with solution for
as ,
at ,
For : with solution
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Series solution for Laplace equation in polar coordinates
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Construct series
Find constants , from Dirichlet B.C.
Apply orthogonal Fourier series formulas
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Cylindrical coordinates
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Gradient of scalar function
Divergence of vector function
Laplacian of scalar function
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Laplace equation in cylindrical coordinates
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Dirichlet problem: find in a cylinder of radius , height
Separation of variables:
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Two stages of separation of variables
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Separation of variables:
Remaining equation
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Series solution
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Series solution for finite length cylinder with constant Dirichlet conditions on cylinder faces
Series solution for infinite cylinder
General series solution for finite length cylinder