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  • 12.4E: Laplace's Equation in Polar Coordinates (Exercises)
    https://math.libretexts.org/Bookshelves/Differential_Equations/Elementary_Differential_Equations_with_Boundary_Value_Problems_(Trench)/12%3A_Fourier_Solutions_of_Partial_Differential_Equations/12.04%3A_Laplace's_Equation_in_Polar_Coordinates/12.4E%3A_Laplace's_Equation_in_Polar_Coordinates_(Exercises)
    \[\begin{array}{c} \ u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta\theta}=0,\quad \rho_0<r<\rho,\quad 0<\theta<\gamma,\\[4pt] u(\rho_0,\theta)=0,\quad u_r(\rho,\theta)=g(\theta),\quad 0\le\theta\le\gam... urr+1rur+1r2uθθ=0,ρ0<r<ρ,0<θ<γ,u(ρ0,θ)=0,ur(ρ,θ)=g(θ),0≤θ≤γ,uθ(r,0)=0,uθ(r,γ)=0,ρ0<r<ρ,
  • 12.3: Laplace's Equation in Rectangular Coordinates
    https://math.libretexts.org/Bookshelves/Differential_Equations/Elementary_Differential_Equations_with_Boundary_Value_Problems_(Trench)/12%3A_Fourier_Solutions_of_Partial_Differential_Equations/12.03%3A_Laplace's_Equation_in_Rectangular_Coordinates
    This section deals with a partial differential equation that arises in steady state problems of heat conduction and potential theory.

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