http://math.stackexchange.com/questions/811901/understanding-the-laplace-operator-conceptually
I think the most basic thing to know about the Laplacian
Notice that the integration by parts formula can be interpreted as telling us that
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The Laplace operator: those of you who now understand it, how would you explain what it "does" conceptually? How do you wish you had been taught it?
Any good essays (combining both history and conceptual understanding) on the Laplace operator, and its subsequent variations (e.g. Laplace-Bertrami) that you would highly recommend? | |||||||||||||
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The Laplacian
and integrate: The integrals where The Laplace-Beltrami operator is essentially the same thing in the more general Riemannian setting - all the nasty curvy terms will be higher order, so the same formula should hold. | |||||||||||||
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I think the most important property of the Laplace operator
Some good books on the subject:
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To gain some (very rough) intuition for the Laplacian, I think it's helpful to think of the Laplacian on
Just as Anthony's answer discusses, the second derivative at Generally, a function is harmonic if and only if it satisfies the mean value property. In The maximum principle states roughly that if Finally, let me go up one dimension and mention some of my intuition for harmonic functions | |||
Another view along the lines of the answers above:
Suppose you have some region in the plane What does "as smooth as possible" mean? Well, one measure of the smoothness of How do we minimize
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Here is some intuition:
I think the most basic thing to know about the Laplacian Notice that the integration by parts formula can be interpreted as telling us that |