Recent content by Dunkle

Homework Statement This is from Chapter 11 of Artin's Algebra: Let p be a prime, and let A (not the identity) be an nxn integer matrix such that A^{p}=I. Prove that n \geq p-1. Homework Equations This is in the factorization chapter, and the section is called Explicit Factorization of...

That was the approach I was attempting to take, but I couldn't figure out to show this explicitly. I think the key here is that all the diagonal entries of U are 1. It sort of makes sense that the algorithm I used in part b will give a unique upper triangular matrix, and then one would simply...

Homework Statement Most invertible matrices can be written as a product A=LU of a lower triangular matrix L and an upper triangular matrix U, where in addition all diagonal entries of U are 1. a. Prove uniqueness, that is, prove that there is at most one way to write A as a product. b...

What is the limit as k goes to infinity of \frac{k(k+2)}{(k+3)^{2}} ? What does this tell you?

I am having problems with the induction for the derivatives. I already showed f '(0)=0. Assuming f^{(k)}(0)=0, we need to show that f^{(k+1)}(0)=0. We compute f^{(k+1)}(0)=\frac{f^{(k)}(x)-f^{(k)}(0)}{x-0}=\frac{f^{(k)}(x)}{x} But I don't see why this is 0. Any suggestions?

Homework Statement This is a claim from a Wikipedia page about analytic functions (http://en.wikipedia.org/wiki/Analytic_function), and I can't seem to prove it. If (r_{n}) is a sequence of distinct numbers such that f(r_{n}) = 0 for all n and this sequence converges to a point r in the...

I think what the question is asking is the following. If you are given some finite group, is it isomorphic to the Galois group of some polynomial in Q?

Split up the last fraction so that you have \frac{2x}{x^{2}+4} + \frac{1}{x^{2}+4}. Use a simple substitution for the first fraction and then the second fraction will involve the inverse tangent. In general, \int \frac{dx}{x^2+a^2} = \frac{1}{a} arctan(\frac{x}{a}) + C

First, you need to parameterize the surface, remembering that a complete parameterization has bounds for the parameters. One parametrization is: \vec{r}(x,y) = <x, y, \sqrt{a^{2}-x^{2}-y^{2}}> -a \leq x \leq a, -\sqrt{a^{2}-x^{2}} \leq y \leq \sqrt{a^{2}-x^{2}} Do you see how this...

Your integral should be either \sqrt{3}\int_0^1 \int_0^{1-x} 1-y \ dydx \ \ \ \ or \ \ \ \ \sqrt{3}\int_0^1 \int_0^{1-y} 1-y \ dxdy I think the second is easier to integrate, so let's do it! \sqrt{3}\int_0^1 \int_0^{1-y} 1-y \ dxdy = \sqrt{3}\int_0^1 \left[x-xy\right]_0^{1-y} \ dy...

The intercepts are indeed fixed. It would also be perfectly fine to let y go from 0 to 1 and then let x go from 0 to 1-y if we switch the dydx to dxdy. Maybe this will help: Draw out the region in the xy-plane. Now, we know the bounds for the outermost integral must be constants (or else we...

Wouldn't your y bounds have to be from 0 to 1-x? Edit: In the xy-plane, the "shadow" of the plane x+y+z = 1 will look like the triangle bounded by x=0, y=0 and x+y = 1. Thus, if you say x is bounded by 0 and 1, then when you hold an arbitrary x constant, y is bounded by 0 and 1-x. I worked out...

It seems like "approach path" would mean the quickest route for the shark. In other words, the shark would have to follow the gradient vector at each point because that is when the directional derivative is as large as possible.

You forgot to divide both sides of the equation by the leading term! Your ODE in standard form should be y' + (4t^{-1})y = e^{-t}t^{-3} The good news is that you found the correct integrating factor and you now have an easier integral to evaluate!

Yes, Hurkyl, I meant there exists an integer k > 1. Sorry about that.