11 Separation in Polar, Cylindrical and Spherical Coordinates
Drums are round, pipes are cylinders, and planets are spheres. In such regions the boundary is a coordinate surface of polar, cylindrical or spherical coordinates, and separation of variables in those coordinates produces new ordinary differential equations in the radial and angular variables, whose solutions are the special functions of mathematical physics: Bessel functions for circles and cylinders, Legendre functions and spherical harmonics for spheres. Their series solutions and basic identities belong to the theory of ordinary differential equations; this section uses them to solve the equations of the previous sections in round regions.
11.1 The Laplacian in curvilinear coordinates
11.2 The vibrating drum
11.3 Heat in a cylinder
11.4 Spherical harmonics and Legendre polynomials
11.5 The age of the Earth
11.6 Special functions in software
11.7 Practice Problems
11.2 The vibrating drum
11.3 Heat in a cylinder
11.4 Spherical harmonics and Legendre polynomials
11.5 The age of the Earth
11.6 Special functions in software
11.7 Practice Problems
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