Recent content by Arkuski

I want to show that if G is a smooth manifold and the multiplication map m:G×G\rightarrow G defined by m(g,h)=gh is smooth, then G is a Lie group. All there is to show is that the inverse map i(g)=g^{-1} is also a smooth map. We can consider a map F:G×G\rightarrow G×G where F(g,h)=(g,gh) and...

I'm trying to come up with a simple proof that if M is an embedded submanifold of N, and P is an embedded submanifold of Q, then M×P is an embedded submanifold of N×Q. I'm thinking this could be easily done using showing that M×P satisfies the local k-slice condition, or that the product of...

Euler's method is a recursive method. In general, it is best to understand what you're actually doing than to memorize the equations. In Euler's method, you essentially get an estimate of y' and use that to estimate y at future values. At t=0, y=1 (given), and thus, y'(0)=3+0-1=2. In the next h...

It depends on the bounds that the integral has. Let c\in ℝ^+. The integral \displaystyle\int_c^∞\frac{1}{x^p}dx is convergent if p>1 and divergent otherwise. However, the integral \displaystyle\int_0^c\frac{1}{x^p}dx is convergent if p<1 and divergent otherwise. Please note that in the...

Yes you have the general idea. In a sense, we're sandwiching the integral \displaystyle\int ^{\pi}_0 \frac{\sin ^2x}{\sqrt{x}} between two integrals with finite value, showing that it converges. You have the upper bound right; since \sin ^2x\le 1, then \displaystyle\int ^{\pi}_0 \frac{\sin...

My mistake, I meant S^1×ℝ instead of M×N

Show that TS^1 is diffeomorphic to TM×TN. (TS^1 is the tangent bundle of the 1-sphere.) We can use the theorem stating the following. If M is a smooth n-manifold with or without boundary, and M can be covered by a single smooth chart, then TM is diffeomorphic to M×ℝ^n. Clearly, I must be...

Thanks for the help guys!

I have a homework problem discussing the heat equation, and I have two questions about notation that I could not figure out through google searching: L^1(ℝ) as in a function g\in C^1(ℝ)\cap L^1(ℝ) ||u(\bullet ,t)||_{L^∞} The bullet in the second expression is actually a single dot at that...

It honestly depends on what you want to do. As a point of reference, it is often stated that John Von Neumann (1903-1957) was the last person to know all of mathematics. It is likely that no one in the future will ever be able to have such a significant impact on mathematics as figures as Euler...

Sorry guys, I have some differential topology homework, and I may be asking a lot of questions in the next few days. Problem Statement Suppose M_1,...,M_k are smooth manifolds and N is a smooth manifold with boundary. Then M_1×..×M_k×N is a smooth manifold with a boundary. Attempt Since...

Give an example of a sequence \{ f_n\} of continuous functions defined on [0,1] such that \{ f_n\} converges pointwise to the zero function on [0,1], but the sequence \{ \int^{1}_{0} f_n\} is unbounded. I'm pretty lost on this one.

Prove that the series \displaystyle\sum_{k=1}^{\infty}\sqrt[k]{k+1}-1 diverges. I thought that I could show the n^{th} term was greater than \frac{1}{n} but this is turning out to be more difficult than I imagined. Is there a neat proof that n^n>(n+1)^{n-1}?

Suppose that g:[a,b]\rightarrow[c,d] is Riemann integrable on [a,b] and f:[c,d]\rightarrow ℝ is Riemann integrable on [c,d]. Prove that f\circ g is Riemann integrable on [a,b] if either f or g is a step function. The proof for g being a step function seems easy enough, but the other way seems...

Mute I fixed it. It was flipped