What is the laplace operator?

The laplace operator is the divergence of the gradient of a function.

The gradient is defined as a vector operator that results in a vector that points in the direction of greatest rate of change.

The divergence is a vector operator that results in a magnitude (scalar quantity) of the outward flux density around a point in space.

So, because the laplace operator is the divergence of a vector quantity, its result is a scalar whose magnitude is proportional to the maximum rate of change.

If the gradient of a scalar field is nonzero, there exists a vector field that points in the direction of the maximum rate of change. Taking the divergence at a point in this vector field, one obtains a scalar whose magnitude represents the greatest rate of change at that point. If the magnitude of this value (the laplacian of the scalar field) is positive, the maximum rate of change is occurring in an outward direction from that point. If it is negative, the maximum rate of change is occurring in an inward direction toward that point.