Recent content by leach

Yes, I think I misunderstood the question. Assuming that we make the hypothesis of constant angle, we find two possible trajectories for the dog, both straight lines. One of them is y=\frac{16+\sqrt{91}}{9} x, which you mentioned earlier, and the other is a divergent trajectory.

I read elsewhere that the discoverer of Stokes theorem was Henri Cartan, Elie's Cartan son. Henri Cartan was member of the Bourbaki group. I agree. These two theorems enclose the true beauty of calculus. I think that the derivation of Euler's variational equation is other good example of that...

I've tried to solve the problem, but the computations are rather complicated. I've found a reference about this problem in http://mathworld.wolfram.com/PursuitCurve.html" . The problem seems to be that the dog is twice as fast as the frisbee, and this prevents a fortunate simplification in the...

The linear (homogeneous) functions in one variable, 'r' in this case, have the general form: f(r) \ = \ \alpha\cdot r,\quad g(r) \ = \ \beta\cdot r, \qquad \alpha,\beta\in\mathbb{R} So this system can be trivially solved, since the variables are separated. On one hand, you have...

You may consider the line integral as the complex path integral \int_\gamma\, \frac{f'(z)}{f(z)-a}\,dz where \gamma(t) = c + it, for -d\le t\le d. Since the integrand has the trivial primitive G(z) = \ln(f(z)-a), you may indeed consider that: \int_\gamma\, \frac{f'(z)}{f(z)-a}\,dz...

That theorem is obviously wrong. The function f(z) = z^{k+1} is a counterexample. It verifies f^{k}(0) = 0, is holomorphic in any domain, yet it is nonzero for every z\ne 0. Most probably your theorem is one of the following: 1) If f^k(z_0) = 0 for all k\ge 0, then f=0 in D. 2) If f=0...

A very interesting problem. I'm not very fluent with Galois theory these days, so please take my words with a grain of salt. For the 7th roots of unity we have the polynomial (x-1)^7 \quad = \quad (x-1)(x^6+x^5+x^4+x^3+x^2+x+1) So the interesting polynomial for these roots is: p...

As you know, the square root has two branches in the real numbers: the positive roots and the negative roots. Positive roots and negative roots are separated by the square root at x=0, which is the ramification point. You can prolongate any branch of the square root to x=0, just defining...

I would try converting the euclidean ball to the infinity norm ball. That is, take a point in the circle, i.e., a vector (x,y) such that x^2+y^2 = 1. Now take the map: f(x,y)\; := \; \frac{(x,y)}{\mathrm{max}\{\vert x\vert,\,\vert y\vert\}} where f(0,0) is undefined. This map takes...

The general case of Stokes Theorem was the first great publication by Nicolas Bourbaki. Trying http://www.google.com/search?hl=en&q=bourbaki+%22stokes+theorem" with bourbaki and "stokes theorem" gives some good references.