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The LAPLACIAN Operator

If we form the scalar ("dot") product of $\Grad{}$ with itself we get a scalar second derivative operator called the LAPLACIAN:

$$\Grad{} \cdot \Grad{} \; \equiv \; \Delsq{} \; \equiv \; {\ . . . . . . \over \partial y^2} \, + \, {\partial^2 \over \partial z^2}$$

What does the $\Delsq{}$ operator "mean?" It is the three-dimensional generalization of the one-dimensional CURVATURE operator d²/dx². Consider the familiar one-dimensional function  h(x)  where  h  is the height of a hill at horizontal position x. Then  dh/dx  is the slope of the hill and  d²h/dx²  is its curvature (the rate of change of the slope with position). This property appears in every form of the WAVE EQUATION. In three dimensions, a nice visualization is harder (there is no extra dimension "into which to curve") but $\Delsq{\phi}$ represents the equivalent property of a scalar function $\phi(x,y,z)$.