Talk:Advanced Mathematics for Engineers and Scientists/The Laplacian and Laplace's Equation



The neighborhood of some point is defined as the open set that lies within some Euclidean distance δ from the point. Referring to the picture at right (a 3D example), the neighborhood of the point $$(x_0, y_0, z_0)$$ is the shaded region which satisfies:


 * $$(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 < \delta^2 \,$$

The Laplacian being zero implies a test point is the average of its neighbours, but not vice-versa. Any function with well behaved Taylor expansion will exhibit linear behaviour in the closest neighbourhood of a test point. In the Taylor's expansion, the linear term will dominate, as the step size h approaches zero: all higher order terms in h vanish, and the test point becomes an average of its neighbours, as they are all linearly related.

To achieve a potent view, the notion of a point being close to the average of its neighbours needs to be augmented with the condition that dividing the 2nd difference by h squared leads to zero as h goes to zero.

Example: The potential around a charged sphere in an otherwise empty universe:

This is v ~= r to the minus one, the solution of Poisson's equation. All functions of the form r to the minus any integer k, will satisfy "the function is the average of its close surrounding neighbours" but only k = -1 satisfies Poisson's equation.