Recent content by timon

Homework Statement A crystal contains N atoms which posses spin 1 and magnetic moment \mu. Placed in a uniform magnetic field B the atoms can orient themselves in three directions: parallel, perpendicular, and antiparallel to the field. If the crystal is in thermal equilibrium at temperature...

I think I am confused by the notation for intervals. For any finite numbers a and b , (a,b) is an open interval, and every closed interval can be written [c,d], where c and d are its endpoints. But this isn't true for unbounded intervals? In other words, (-\infty, \infty) is a closed...

Good day dear fellows. I am given the following series h(x) = \sum_{n=1}^{\infty} \frac{1}{x^2+n^2}. It is requested that I show that h(x) is continuous on R. I did the following: use the Weirerstrass M-test to show uniform convergence, and then, using the continuity of the functions...

thanks a lot! I tried to do the same thing but couldn't get g_i - g_{i+1} to converge. I feel somewhat silly now!

Homework Statement This is a homework question for a introductory course in analysis. given that a) the partial sums of f_n are uniformly bounded, b) g_1 \geq g_2 \geq ... \geq 0, c) g_n \rightarrow 0 uniformly, prove that \sum_{n=1}^{\infty} f_n g_n converges uniformly (the whole...

It is indeed quite amazing that I managed to screw the same trivial manipulation up three times in a row, always in the same manner. Thank you for the help.

Thanks for the quick reply. That actually wasn't a typo. I managed to switch the indices the same way in three separate calculations, getting the same erroneous result each time. Are these the correct eigenvectors? \frac{1}{\sqrt{2}} \left[ \begin{array}{c} 1 \\ i \\ \end{array} \right] ...

Homework Statement This question is from Principles of Quantum Mechanics by R. Shankar. Given the operator (matrix) \Omega with eigenvalues e^{i\theta} and e^{-i\theta} , I am told to find the corresponding eigenvectors.Homework Equations \Omega = \left[ \begin{array}{cc} \cos{\theta}...

1. The problem statement let \vec x and \vec y be linearly independent vectors in R^n and let S=\text{span}(\vect x, \vect y). Define the matrix A as A=\vec x \vec y^T + \vec y \vec x^T. Show that N(A)=S^{\bot}. 2.equations I have a theorem that says N(A) = R(A^T)^{\bot}. A is symmetric; A...

tip 1: If the frequency changes, the color of the light changes. tip 2: the speed of light, c, is constant under all circumstances. tip 3: \lambda = \frac{c}{f}

o wait i see what I've been doing. x and y are both on the x axis, so x=y doesn't imply x=y=0, in fact x and y could be any real number. but the equality is still true, and trivially so. I guess I'm done? if so, thanks a lot for the assistance!

sec(x)[2tan(x)+sec(x)] = 0 You are trying to find an x to make the left side equal to 0. Like you said, there is no x for which sec(x) = 0, so you can discard that possibility. How else can the left side of that equation be 0?

i used the mean value theorem, as you suggested: if f(x) = arctan(x) then \exists c such that f(c)' = \frac{f(y)-f(x)}{y-x} \Leftrightarrow f(c)'(y-x)=f(y)-f(x) for any x, y \in \Re where x < y, since arctan is continuous and differentiable everywhere. also, f(x)'=...

is this correct? x-y > 0 \Rightarrow x - y = |x-y|, arctan(x)-arctan(y) = |arctan(x)-arctan(y)| \Rightarrow |x-y| \geq |arctan(x)-arctan(y)| x-y < 0 \Rightarrow x - y = -|x-y|, arctan(x)-arctan(y) = -|arctan(x)-arctan(y)| \Rightarrow -|x-y| \geq -|arctan(x)-arctan(y)| \rightarrow |x-y|...

i can get arctan(x) - arctan(y) \leq x-y, but how do i get that to say something about the absolute value?