Recent content by exclamationmarkX10

Maybe that is what Spivak meant when he says "Precisely the same considerations hold for forms" on page 116. I see it now, thanks for all your help Orodruin.

Taking the suggested path, I reduced the problem to showing for all j, d(\omega_{i_1, \ldots, i_p} \circ g)(b)(w_j^{\prime}) = d(\omega_{i_1, \ldots, i_p} \circ f)(a)(w_j) where \omega_{i_1, \ldots, i_p} are components of \omega, g: V \rightarrow \mathbb{R}^n is another coordinate system around...

I am trying to finish the last chapter of Spivak's Calculus on Manifolds book. I am stuck in trying to understand something that seems like it's supposed to be trivial but I can't figure it out. Suppose M is a manifold and \omega is a p-form on M. If f: W \rightarrow \mathbb{R}^n is a...

The maximum value of f(z) doesn't make sense since f(z) is a complex number. In general, can one complex number be considered less than another complex number? e.g. is 3i "less than" 4i? These numbers cannot be compared this way however real numbers can. Have you learned the maximum modulus...

The solution given to you says you should get something with x in it after taking limits? If you can find a value for the integral, then you know it converges. Taking the limit of the integrand as x goes to infinity will only tell you whether or not the integrand goes to zero. In this case...

After that, you should get the same equations of motion except they are multiplied by f\prime(L). You have to then argue that you can divide out the f\prime(L).

Yes, you could've started out that way. But you have to keep in mind that there can be more than one microstate for each speed so that the probability might be proportional to more factors that contain v. For each infinitesimal interval of speed (v, v + dv), there is an entire spherical shell...

Substitute the lagrangian with f in the euler-lagrange equations. Then use chainrule.

Thanks for clearing that up.

With your first suggestion, how do you get rid of the nagging constant that results from finding the equilibrium solution to the pde with generation term? Also, with your second suggestion, don't you have to assume that the solution is C^2 and that f(x) is differentiable? I think the series...

Actually, I'm not even sure if (b) can be proven with the info given because proving that the limit of \theta as h goes to 0 might be a problem. You can at least say: if the limit of \theta exists, it is 1/2.

For part (a), you should try l'hopital's rule directly for the left side of the equation. For part (b), you can use your first attempt's last line and do the trick: multiply by \theta / \theta like gopher_p mentioned and use the fact that f''(x_0) \neq 0 to show (b) (you need this fact because...

No, you can't just let k = 0 (try going through your arguments with x=12 and y=0). Remember what you are trying to show: You want to show that the congruence classes of x and y are equal, not x=y.

My mistake, I thought you were saying \sup P=0

Use the squeeze theorem. i.e. find another function g that is greater than or equal to your function for positive x values and \lim_{x \to +\infty} g(x) = 0.