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'.
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...
Comments
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...