Today I want to run through an example of how we use our new tools to read geometric information out of a parallelogram. I’ll work within with an orthonormal basis and an identified origin to give us a system of coordinates. That is, given the...

So, why bother with this orientation stuff, anyway? We’ve got an inner product on spaces of antisymmetric tensors, and that should give us a concept of length. Why can’t we just calculate the size of a parallelepiped by sticking it into this bilinear...

Yesterday we established that the -dimensional volume of a parallelepiped with sides should be an alternating multilinear functional of those sides. But now we want to investigate which one. The universal property of spaces of antisymmetric tensors...

And we’re back with more of what Mr. Martinez of Harvard’s Medical School assures me is onanism of the highest caliber. I’m sure he, too, blames me for not curing cancer. Coming up in our study of calculus in higher dimensions we’ll need to...

On Tuesday I posted a still from The Coen Brothers’ new film A Serious Man. Firmly rooting the film in mid-1967, and in a sense of existential crisis and alienation, is the repeated use of Jefferson Airplane’s hit, “Somebody To Love”. When the truth...