Trying to visualize the projection map using fibers. You'll have to take my word that the lines stop before getting to the origin.

Today’s Theorem of the Day (TM) I used to compute the Jacobian of a radial projection. In particular, consider the map where for all . This projects all of *n*-space onto the surface of the unit ball, and leaves the interior untouched. Then we may compute the derivative .

To calculate the Jacobian of *F *means we have to calculate the determinant of that matrix. With a little figuring, we can write that last sentence as .

Now we apply The Theorem, which Terry Tao quoted Percy Deift as calling (half-jokingly) “the most important identity in mathematics”, and wikipedia calls, less impressively, “Sylvester’s determinant formula“. Its usefulness derives from turning the computation of a very large determinant into a much smaller determinant. At the extreme, we apply the formula to vectors *u* and *v*, and it says that . In our case, it yields . Thus we turned the problem of calculating the determinant of an *n x n *matrix into calculating the determinant of a 1 *x *1 matrix.

Pretty nifty.

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