The Geometrical Significance of the Laplacian
The Laplacian operator can be defined, not only as a differential operator, but also through its averaging properties. Such a definition lends geometric significance to the operator: a large Laplacian at a point reflects a “nonconformist” (i.e., different from average) character for the function there. This point of view is used to motivate the wave equation for a drumhead.
Styer, Daniel F. 2015. "The Geometrical Significance of the Laplacian." American Journal of Physics 83(12): 992-997.
American Association of Physics Teachers
American Journal of Physics
Physics and Astronomy
Kinematics, Laplace equations, Poisson's equation, Wave equations, Fluid equations