Math PhD, currently working as a software engineer.
PatrikN
Karma: 18
Thinking about how to prove the multilinearity of the volume of a parallelepiped definition I like this sketched approach:
The two dimensional case is a “cute” problem involving rearranging triangles and ordinary areas (or you solve this case in any other way you want). The general case then follows from linearity of integrals (you get the higher dimensional cases by integrating the two dimensional case appropriately).
Nice theorem and write up. Already the one dimensional case is something interesting called “The second symmetric derivative”. And I think it might be used to prove the general case directly: If you add up that result for n orthogonal directions then the left hand side is the Laplacian. The right hand side is a sum of a n limits that at first glance seems to depend heavily on the picked direction, but they can’t as the left hand side is independent of the picked directions. We are free to pick arbitrary many different directions and take the average. In the limit it becomes the average over a sphere but the order of limits is wrong, some standard theorem might apply to resolve that?