Hello,
I am looking at the proof (Theorem 2.5 (b) Apostol) of $$ \phi (mn) = \phi(m) \phi(n) \frac{d}{\phi(d)} $$ where $$ d = (m, n) $$.
Can someone explain how they go from
$$ \prod_{p|mn} \left( 1 - \frac{1}{p} \right) = \frac{\prod_{p|m} \left( 1 - \frac{1}{p} \right) \prod_{p|n}...
I've followed my link above and I think your limits for z are r < z < 1/√2
My triple integral is
int (0 to 1/√2) r dr int (0 to 2∏) dθ int (r to 1/√2) dz
Hi oli4. Thanks for the swift reply
There are a few things I don't understand
1. Where did they get n = a +b from. Why is that important?
2. When they say assume that the theorem has been proved up to n-1, why do they choose n -1
3. Why do they apply the theorem to (a-b) and (b). I don't...
Hello to all,
Wow this thread has really taken off. Thanks for all the suggestions. I actually bought Velleman's book. I just finished the first three chapters. It's really well written.
Good luck.
Hello,
I working through my notes on Nother's theorem and invariance under translation. I don't understand how they get expression 7.3, from the line before 7.3 (see attachment). Can anyone explain.
Thanks.
Well ordinarily you have a function y(x), where x is the independent variable and y is the dependent variable.
With parametric derivatives (of x(t) and y(t) let's say, which depend on t) you have x'(t) and y'(t) . Here x and y are dependent and a function of independent variable t.
So I...
Homework Statement
I have a problem
u'' + lambda u = 0
with BCs: u'(0) = b*u'(pi), u(0) = u(pi).
where b is a constant.
I have to find b which makes the BCs and problem self-adjoint.
Homework Equations
see below
The Attempt at a Solution
I see in my notes...
THomework Statement
Solve x*y'' + y' - a*y = 0
where a > 0
Homework Equations
Not sure what's relevant here. See Below.
The Attempt at a Solution
I think this can be solved by changing the independent variable. I tried x = √t, x = 1/t, x = ln(t), x = exp(t) but these seem to...