The Laplacian Operator
The Laplacian operator, writtenis probably the most important operator in mathematical physics. It allows us to compare the value of a function at a point with the value of the function at neighbouring points:
Ifat a pointthenis smaller than the average ofat nearby points, say on a circle centred at
Ifthenis equal t6 the average value ofat neighbouring points.
Ifat a pointthenis greater than the average ofat nearby points..
The heat equationmeasures temperature or concentration and can be interpreted to mean that the temperature at a point is increasingifso the temperature at that point is smaller than the temperature at neighbouring points.
The wave equationdescribes the acceleration of a string or drumhead and can be interpreted to mean that the acceleration of the drumheadis proportional toThat is, the string or drumhead is accelerating up if the displacement of the string or drumhead is less than neighbouring points.
Laplace's equationsays that the solutionis equal to the average ofat neighbouring points.
Poisson's equationcan describe a range of phenomena. Ifwherecharge density, then describes the electrostatic potential, Ifrepresents a heat source at the pointthendescribes the temperature at
Helmholtz's equation(reminiscent of the equation for simple harmonic motion) describes the motion of a stretched membrane.
We can write the Laplacian operator in one, two or three dimensions, in cartesian, polar, spherical or cylindrical coordinates depending on the problem to be solved.