Cartesian Coordinates In Cartesian coordinates, Laplace’s equation  S(x,
y, z) = 0 is written as
 where −∞ < x < ∞, −∞ < y < ∞, −∞ < z < ∞ . Applying separation of variables we assume a product solution  Substituting this form of S into Laplace's equation and dividing by S gives  Each term in the above equation must be equal to a constant if the sum is zero for all x, y, and z, since these variables may vary independently  where α2 + β2 = γ2 . The solutions to these differential equations may be written as linear combinations of sine, cosine, and exponential functions so that we may construct a product solution  with constants determined by the boundary conditions. As an example of the application of separation of variables, consider a unit square region with boundary condition S = 0 on three sides and S = 1 on the other side. Since there is no z dependence α2 + β2 equation (2.5) become  Applying the boundary conditions S(0,y) = S(1,y) = 0, where the sine function is zero given α = nπ and relabeling the constants and writing the solution as a sum  The condition S(x, 1) = 0 gives Bn = -An cosh(nπ)/sinh(nπ) and  Finally, Anis determined by applying the boundary condition at S(x, 1) = 1  Multiplying both sides of this expression by sin(nπx) and integrating  Making use of orthogonality, where only terms with n = n′ remain, and using the identity sin2(nπ) = (1 − cos 2nπ)/2, we have  where Anis zero for even n so the sum is over odd n  The following MATLAB program calculates equation (2.12) and plots the solution over the unit square:  In the following coordinate systems we consider Laplace's equation with axial symmetry and no variation in the φ-direction, or with planar symmetry and no variation in the z-direction. These results will be useful for comparison to QuickField problems that have either axial or x-y symmetry. Cylindrical Coordinates In cylindrical coordinates with axial symmetry, Laplace's equation S(r, z) = 0 is written as  where r ≥ 0, −∞ < z < ∞. Substituting S(r, z) = R(r)Z(z) with separation constant k2 gives the differential equations  so that we may construct our solution  where J0(kr) and N0(kr) are Bessel functions of zero order. Laplace's equation in cylindrical coordinates without variation in the z-direction S(r,φ) = 0 is written as 
where r ≥ 0, 0 < φ ≤ 2π. Substituting S(r,φ) = R(r)Φ(φ) 
with terms that may include a0ln(r) + b0φ + c0, where a0, b0and c0 are constants. Bicylindrical Coordinates The Laplacian S(α, β) is written in bicylindrical coordinates  where −∞ < α < ∞, 0 ≤ β ≤ 2π. Substituting S(α, β) = Λ(α)Ω(β) with separation constant p2 gives  Elliptic Cylindrical Coordinates The Laplacian S(α, β) =0 is written in elliptic cylindrical coordinates  where 0 ≤ α, 0 ≤ β ≤ 2π. Substituting S(α, β) = Λ(α)Ω(β) with separation constant p2gives  Spherical Coordinates In spherical coordinates with axial symmetry, Laplace's equation S(r,θ) = 0 is written as  Substituting S(r,θ) = R(r)Θ(θ) with separation constant l(l + 1) we obtain  where Pland Qlare Legendre functions of the first and second kind, respectively. As an example of a spherically symmetric solution to (2.22) consider a spherical surface of radius R with boundary conditions S(R,θ) = V0and S(r → ∞,θ) = 0. Since there is no θ dependence we take l = 0 so that (2.23) becomes  The boundary condition at infinity gives A = 0. The boundary condition at r = R gives B = RV0and we have  Note that the same result may be obtained by setting the radial part of (2.22) equal to zero and integrating twice. Prolate Spheroidal Coordinates Laplace's equation in prolate spheroidal coordinates with axial symmetry S(ξ,η) = 0 is written as  where ξ ≥ 0, 0 ≤ η ≤ π.. Substituting S(ξ,η) = Y(ξ)Θ(η) with separation constant l(l+1) gives  We now consider a football shaped prolate spheroidal surface ξ = ξ0with potential V0. There is no η dependence so that (2.27) becomes  where P0(x) = 1. Applying the boundary conditions S(ξ → ∞) = 0 and S(ξ → ξ0) = V0 we have  This is equivalent to  which may also be obtained by setting the first term of (2.26) equal to zero and integrating twice. Fortified by this simple result it might be tempting to find the potential between two opposing electrodes in the shape of hyperboloids of revolution by setting the second term in (2.26) equal to zero and integrating twice. However, this procedure would not yield a solution over the region between the electrodes, where £ is varying. Instead we must use (2.27)  where the Ql(cosh ξ) are divergent at ξ = 0 and are discarded. The constants Aland Blare determined by boundary conditions at η = η0, making use of the orthogonality of Pl(cosh η) and Ql(cosh η). Oblate Spheroidal Coordinates Laplace's equation in oblate spheroidal coordinates with axial symmetry S (ξ,η) = 0 is written as  where ξ ≥ 0, 0 ≤ η ≤ π. Substituting S(ξ,η) = Υ(ξ)Θ(η) with separation constant l(l+1) gives  Toroidal Coordinates The Laplacian S(α, β) is written in toroidal coordinates as  where α ≥ 0, −π ≤ β < π.. Substituting S(α, β) = Υ(α)Θ(β) with separation constant v2gives  | 
