Recent content by MissMoneypenny

Thanks a lot for your suggestions. I'll head to the library and have a look at those books.

I've been trying to self study the section on de Rham cohomology in Guillemin and Pollack's book Differential Topology. The section is in a sense hands on: most of the results are presented as exercises scattered throughout the section, and some hints are given. I've hit a road block in a few of...

Thanks for the help. I followed your advice and worked out similar formulas for m! and for (pk-m)!, and managed to prove the result by bounding the difference using geometric series. I appreciate your comment, it gave me the little push in the right direction that I needed.

Homework Statement Let p be a prime, k be positive integer, and m ∈ {1, 2, 3, ..., pk-1}. Without using Lucas' theorem, prove that p divides \binom{p^k}{m}. Homework Equations The definition of the binomial coefficients: \binom{a}{b} = \frac{a!}{b! (a-b)!} The Attempt at a Solution I've...

You're correct, what I wrote is a strange problem statement. It was my fault. The full problem was to show S is a subfield of C. However, I knew how to prove all of the other field axioms hold for S, so I changed the problem statement. As you correctly pointed out, I didn't change it very...

Thanks for the quick response! I think I see what you're getting at. If I can prove what you wrote, then since each element of S is a linear combination of 1, α, and α2 whose coefficients are in \mathbb{Q}, it follows that each element of S has an inverse in S. Unfortunately I really am not sure...

Homework Statement Let S={p+qα+rα2 : p, q, r \in \mathbb{Q}}, where α=\sqrt[3]{2}. Then S is a subfield of \mathbb{C}. Prove that each nonzero element of S has a multiplicative inverse in S. The Attempt at a Solution Let p, q, r\in\mathbb{Q} such that not all of p, q, r are zero. If...

Ah, that makes sense! I think I've got it now and can safely work on my solution, but if I run into any more questions I'll make another post. Thanks again for your help!

Thanks for the help PeroK and Hallsoflvy. So to establish that a and b can be positive can I simply comment that if (a/b)2=2, then it is also true that (-a/b)2=2 and that (-a/(-b))2=2, so it doesn't matter whether we consider the case in which a and b are both positive, both negative, or in...

Homework Statement Prove that √2 is irrational as follows. Assume for a contradiction that there exist integers a, b with b nonzero such that (a/b)2=2. 1. Show that we may assume a, b>0. 2. Observe that if such an expression exists, then there must be one in which b is as small as...

Awesome, thanks a lot for the hint! Following your hint I got the correct answer. Thanks again, have a great day :)

This problem is listed several chapters before the residue theorem is presented, so I think that there must be some way to do it without the residue theorem. On the midterm I have coming up we won't be allowed to use the residue theorem, so I was wondering if there's another way to do it. The...

Hi vela, Thanks for the reply! Yes, there should certainly be limits on the original integral. Integration is performed from 0 to 2π. I've added the limits to my post. You're also correct about my algebra error. Nonetheless, even with the correct integrand I'm not sure where to go. Thanks...

Homework Statement Compute the real integral \int\frac{dθ}{2+sin(θ)}, where the limits of integration are from 0 to 2π by writing the sine function in terms of the exponential function and making the substitution z=e^{iθ} to turn the real integral into a complex integral. Homework...

Hi TSny, Thanks so much for the help! You were completely correct that the torque at θ=π would be in the wrong direction to reverse the direction of the cylinder's rotation. Your hint also managed to give me just enough of a push to see where I went wrong and to determine a new (and physically...