Recent content by feynman137

Thanks, Janus, very informative. I will try to derive the equation above. Calling S the synodic period, t_E the period of Earth and t_M the period of mars, clearly t_E+t=S, where t is the interval of time that is needed after a complete revolution of Earth to catch up again with Mars and theta...

I think it is y=1/x. So, take an arbitrary interval [0,l] and employ riemann sums: divide it in n subintervals, each delta x= l/n wide. The area of all rectangles of basis delta x and height 1/i(delta x), where i runs from 1 to n, is an approximation downwards of the area under the curve. In...

Kepler guessed his third law from a great deal of data about planetary motions that had been taken by astronomer Tycho Brahe. Brahe mostly measured angles and relative distances in the sky without the aid of a telescope. Nonetheless, he was the first to collect precise and reliable information...

My definition of diffeomorphism is a one-to-one mapping f:U->V, such that f and f^{-1} are both continuously differentiable. Now, how to prove that if f is a diffeomorphism between euclidean sets U and V, then U and V must be in spaces with equal dimension (using the implicit function theorem)?

Could you possibly comment the following statement: 'Heat propagation is a semi-deterministic process in that its future is determined by its present but not by its past.' Is heat propagation a violation of the determinacy and reversibility of the laws of classical mechanics? Thanks

I figured it out: just write (e_2 x e_3).(e^2 x e^3) as ((e_2 x e_3) x e^2).e^3 and by basic identities we can prove that VV'=1>0, so the two basis have the same orientation.

If A is invertible, than it is evident that AB and BA are similar matrices, therefore they have the same characteristic polynomial. Otherwise, we notice that the equation in lambda, det(A-lambda I)=0 has finitely many solutions. We can take epsilon such that, for all lambda 0<|lambda|<epsilon...

How to prove that two reciprocal basis are either both right ended or both left-handed? If (e_1,e_2,e_3) and (e^1,e^2,e^3) are two such basis, since the scalar triple products depend on orientation, it would be enough to show that VV'=1 (where V and V' are the volumes, taken with their sign, of...