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This text delves into two-dimensional problems in mathematics, focusing on cylindrical symmetry and conformal mapping. It emphasizes the power and elegance of complex analytical functions, their role in mapping the (x,y) plane onto the (u,v) plane, and their adherence to Cauchy-Riemann equations. The text also discusses the Laplace operator in polar coordinates and provides examples, such as the behavior of potentials in structured systems like two half pipes, equipotential lines, and field lines. It highlights the significance of boundary conditions and analytical functions in solving physical problems.
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3.5 Two dimensional problems • Cylindrical symmetry • Conformal mapping
Example: Two half pipes
Conformal Mapping Is there a simple solution?
iy x Examples: For two-dimensional problems complex analytical function are a powerful tool of much elegance. Maps (x,y) plane onto (u,v) plane. For analytical functions the derivative exists.
Analytical functions obey the Cauchy-Riemann equations which imply that g and h obey the Laplace equation, If g(x,y) fulfills the boundary condition it is the potential. If h(x,y) fulfills the boundary condition it is the potential.
g and h are conjugate. If g=V then g=const gives the equipotentials and h=const gives the field lines, or vice versa. If F(z) is analytical it defines a conformal mapping. A conformal transformation maps a rectangular grid onto a curved grid, where the coordinate lines remain perpendicular. Example w z Cartesian onto polar coordinates: Full plane Polar onto Cartesian coordinates:
equipotentials field lines Edge of a conducting plane
