Instead of creating a new thread, I decided to just write in here.
If we define a set X={p in (0, 1), p in R} and let E = X \ Q.
Now, E non-empty(since Q is countable and X is not) and bounded, furthermore, no point of Q is in E. Now what remains to be shown is that if p in E, then p is a...
HallsofIvy,
I'm studying with Rudin’s Principles of Mathematical Analysis. They define lim sup exactly as sup of a set of all subsequential limits.
So let E be a set of all subsequential limits and let x = lim sup(an) = sup(E), then
if an > x = lim sup(an) for all but finitely many an, it is...
Hi guys. I'm apprently stuck on the basics of the analysis. On the proof that Q lacks least upper boundary property to be precise.
The example I have uses a set A (p in Q | p > 0, p^2 < 2)
then q is defined as p - \frac{p^{2} - 2}{p + 2} . Then they show that if p is in A then q is in A too...
How did you get this expression?
\frac{1+t^2}{(\sqrt{1+a}t^2+2\sqrt{a-b}t+\sqrt{1+a}) (\sqrt{1+a}t^2-2\sqrt{a-b}t+\sqrt{1+a})}
when I factor
{a(1-t^2)^2 + 4bt^2 + (1+t^2)^2} = (a+1)t^4 -2(a-2b-1)t^2 + (a+1)
then say,
u=t^2 and
u1,2 = \frac{(a-2b-1)\pm 2\sqrt{(1+b)(b-1)}}{a+1}...
\int \frac{1}{a*cos^2(\phi) + b*sin^2(\phi) + 1} d\phi
after letting t=\tan \frac{\phi}{2} I get
\int \frac{1+t^2}{a(1-t^2)^2 + 4bt^2 + (1+t^2)^2}
this does not look to facinating, so I let u=t^2 and this simplifies it a bit further, but introduces \sqrt{u} into denominator;
I...
should be r^2 = 2a^2 (cos^2theta);
sketch the curve to find the limits of intergration, I'd say for the right loop you integrate
from -pi/4 to pi/4.
Constant a controls the form of the curve.
Try sketching it here http://graph.seriesmathstudy.com/
Hi,
I'm trying to evaluate this integral over a hemisphere:
\int cos(\theta)^{a*cos^2(\phi) + b*sin^2(\phi)} dw
where dw - solid angle measure,\phi is azimuthal angle and \theta
is polar angle.
Thus we have:
\int cos(\theta)^{a*cos^2(\phi) + b*sin^2(\phi)} dw = \int \int...
This picture show what differential is for a function f(x)
http://www.bymath.com/studyguide/ana/sec/ana4a.gif
Basically, it is the change in the linear approximation for a function for a change in x, dx.
dy/dx = f '(x) -> differential dy = f '(x) dx
When dx is small dy is a good...