Cartesian Coordinates

In Cartesian coordinates, Laplace's equation ? ² S( x,y,z) = 0 is written as

(2.1)

where - ? < x < ?, - ? < y < ?, - ? < z < ?. Applying separation of variables we assume a product solution

(2.2)

Substituting this form of S into Laplace's equation and dividing by S gives

(2.3)

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

(2.4)

where ? ² + ? ² = ? ². 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

(2.5)

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 ? ² = - ? ² equation (2.5) become

(2.6)

Applying the boundary conditions S(0, y) = S(1, y) = 0, where the sine function is zero given ? = n ? and relabeling the constants and writing the solution as a sum

(2.7)

The condition S