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In his answer to an MO question: "Why is the Laplacian ubiquitous?", Terry Tao says that
The Laplacian of a function $u$ at a point $x$ measures the average extent to which the value of $u$ at $x$ deviates from the value of $u$ at nearby points to $x$ (cf. the mean value theorem for harmonic functions).
Would anybody elaborate (probably with examples) how the mean value theorem for harmonic functions relates to the point he makes about the Laplacian?