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## Multivariable calculus

### Course: Multivariable calculus>Unit 2

Lesson 12: Laplacian

# Harmonic Functions

If the Laplacian of a function is zero everywhere, it is called Harmonic. Harmonic functions arise all the time in physics, capturing a certain notion of "stability", whenever one point in space is influenced by its neighbors.

## Want to join the conversation?

• It's worth mentioning that we're not talking about a circle (or disk, to be precise) on the surface itself, but the projection onto the surface of a disk on the xy-plane.
• So, like Grant explains at , you basically taking a circle around a point in the input space (xy plane) and lifting it up to the graph in the z axes (where the output space is).
• What size of the circle (or other boundry of surroundings) are we implying? Is it infinitesimal?
• It's an abstract circle of any size (it doesn't matter what size the circle is, the averege of every value in that circle is alway the original point). Try looking at the graph and making different size circles, you will see it always seems to average equals the selected point
• I don't understand why the neighbouring points are greater on average when the second derivative is positive. Can anyone elaborate on this.
• Positive second derivative indicates that the slope of the first derivative is positive. So, for some positive 'h' value, [f(x+h) - f(x)] will be greater than [f(x) - f(x-h)]. If you average f(x+h), f(x) and f(x-h) values, then the average ( [ f(x)+f(x+h)+f(x-h) ] / 3 ) is always higher than the value of f(x).
• Do you suppose that the curl of the gradient of a function is a useful concept for scalar valued functions? i.e. nabla x nabla f?
• No, it's not since the curl of a gradient of a scalar function is always zero! Try computing it out.
• e^x is a function that e^xplodes!
• In single variable calculus when the second derivative is zero it means that this could be an inflection point or local maxima or local minima. In short, it indicates an "inconclusive result" and that further investigation is required to understand the nature of the curve at that point.
So why is Laplacian being zero such an important result?
• Does the laplacian of a function f(x,y) being equal to zero also mean that given said function f(x,y), and given a circle with center at point (a,b), the volume between the x-y plane and the function f(x,y) inside the circle is equal to the area of the circle multiplied by f(a,b)?
• I don't know but that sounds logical to me!