Recent content by Tantoblin

True, but (1) try giving a sound proof for the triangle inequality, and (2) the law of cosines is usually a consequence of the triangle inequality: i.e. one defines the angle between two vectors as the arccos of X.Y / |X||Y| (which is smaller than one and hence in the domain of arccos).

Isn't this related to the (very hard) Plateau problem?

This is completely off-topic, but when I'm confronted with a bottleneck in Matlab, I always write the critical parts in C and then call it as MEX file. This usually speeds up things by orders of magnitude.

Well, what I wanted to say was (in two dimensions, say), that it doesn't make sense to try to change the PDE U_{xx} - U_{yy} = 0 (hyperbolic) into U_{xx} + U_{yy} = 0 (elliptic). The former has two characteristics while the latter has none. This is related to the theory of quadratic forms in...

I think not. Think of quadratic forms. In three dimensions, can you use a coordinate transform to bring an arbitrary quadratic form into the form \xi_1^2 + \xi_2^2 + \xi_3^2? What quantities are invariant under linear transforms?

Yes, well the crucial point here is that you are applying the open mapping theorem, which works only when a number of conditions are satisfied. The open mapping theorem is very nontrivial and even counterintuitive, so you should properly document its application.

As you pointed out, Jacobi fields measure the separation of geodesics. Suppose you have a bunch of geodesics \gamma_\tau, where \tau is a parameter labelling the individual geodesics. So, we're talking about two-parameter families of curves, and not of diffeomorphisms! Each of these geodesics...

I have the book "A history of mechanics" by Rene Degas. It's usually considered to be very good as far as the early history of mechanics is concerned, but I have no idea how well it treats relativity. It does have a very good treatment of the 19th century discussions on relativity, causality...

For the purposes of visualisation, it is easier to look at the coboundary operator in algebraic topology. In the book "Algebraic Topology" by Allen Hatcher (freely available from http://www.math.cornell.edu/~hatcher/AT/ATpage.html) there is an excellent introduction to the coboundary operator...

If the norm is derived from an inner product (as, it would appear, in your case), there is the following standard geometric argument for the latter inequality. Consider the following self-evident fact: \Vert x + ty \Vert^2 \ge 0, where t is a parameter. Now, the left-hand side is a quadratic...

As for the openness/closedness, z \mapsto z^2 is a holomorphic mapping, and its domain is open and connected, so...

Well, by comparing Taylor expansions you can prove that the sequence you posted is an approximation to the first derivative that is correct to at least second order, contrary to (a_{n+1}-a_{n})/h, which is only first-order. With a little more insight you can convince yourself that any...