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Comment by pixcavator | Hacker News Reader

>If there is an unqiue optimal orientation for a single static panel (which under realistic assumptions there is)...

If you have some kind of “continuous function attains its max” argument here, I think you should elaborate. Because it’s interesting but not really “trivial”.

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No, results specific to solar panel orientation. Math is not trivial at all, but the results are intuitive: roughly, point the panel in the direction which, in mean, is in the direction of the sun. So tilted towards the equator, at the same angle as your latitude.

http://www.nrel.gov/rredc/pvwatts/changing_parameters.html#t...

"The default value is a tilt angle equal to the station's latitude. This normally maximizes annual energy production. Increasing the tilt angle favors energy production in the winter, and decreasing the tilt angle favors energy production in the summer."

(Couldn't find a clear proof of this (under any modelling assumptions), sorry.)

A unique optimum isn't really necessary here (I shouldn't have required it!)

He used the extreme value theorem. The theorem asserts that a continuous function on a compact set (for our purposes closed and bounded set) has a maximum (and a minimum). In this case the the closed and bounded set is the sphere of all possible orientations for solar panels. The theorem is mentioned in most introductory calculus courses, but the proof is definitely not 'trivial'.

http://en.wikipedia.org/wiki/Extreme_value_theorem

That's not actually sufficient to show unique extrema; there an be a set of points on which a function achieves the same, maximal, value.

i.e. cos(x) over x <- [0,4π] -- multiple maxima at {0,2π,4π}.

But we don't really need unique extrema here. Oversight on my part -- there can be (although I believe there aren't) multiple optimal panel orientations. Then the optimal array is an array with each panel having any one of the optimal orientations.

Well, the Calc 1 version isn't good enough because there are several variables here. You have to consider a continuous function defined on the sphere -- to capture all possible orientations of the panel. In fact, it could be a closed subset of the sphere -- to take care of all possible shadows, obstructions, etc. This is nice, you don't see a lot of direct applications of pure existence theorems...

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