Suppose $(X,\mathcal{S},\mu)$ is a measure space such that $\mu(X)<\infty$. Suppose $p,r$ are positive number with $p<r$. Prove that if $f:X\to[0,\infty)$ is an $\mathcal{S}$-measurable function such that $\int\,f^rd\mu<\infty$, then $\int\,f^pd\mu<\infty$.
How this definition of Line Integrals $\int_a^b A(x(t),y(t)) \ x′(t)+B(x(t),y(t)) \ y′(t)dt$ is linked to concept of $F*cos(\theta) * ds$
The only thing that I CAN'T see is the $cos(\theta)$ in the first formula, where is this concept? How the first formula take the tangential component from curve? It's thanks to derivation? If yes, why ?
How this definition of Line Integrals $\int_a^b A(x(t),y(t)) \ x′(t)+B(x(t),y(t)) \ y′(t)dt$ is linked to concept of $F*cos(\theta) * ds$
The only thing that I CAN'T see is the $cos(\theta)$ in the first formula, where is this concept? How the first formula take the tangential component from curve? It's thanks to derivation? If yes, why ?
@geocalc33 If easy, open publication takes too much man power to support (peer review each crackpot entry). Then all proofs should be checked by a computer.
If you understand it, you should be able to write out the basic proofs for yourself, not memorizing and copying. And you must be able to do concrete examples for every concept.
Wait. You don't consult with a faculty member about what you're taking and how you've done in preceding courses? Where I've taught, students have to get the adviser to sign a sheet approving the courses.
You shouldn't just do what "everyone else" is doing.
They can give better advice than I can or we can. If they've taught you they have some idea of what you can do successfully, and might suggest better courses for you.
@TheTerriblePuddle with regards to your question on how to think about generated ideals, if you have a commutative ring with a 1, then a principal ideal, $(r)$ where $r\in R$ is simply $rR$
If you think about $R$ as a module over itself, ideals are submodules, so the situation should be pretty much the same as for vector spaces and vector subspaces: you take a generating set and do finite linear combinations
@The Terrible Note, if you don't have a unity member in the ring (multiplicative identity) then $a$ may not be in $aR$. Do you see why? can you find a counterexample?
@EdwardEvans I'm a bit worried by the fact that Germany is where Italy was around 8-9 days ago as number of cases, but we were already closing down all schools then while Germany doesn't care
also "On 3 March, University of Zurich announced six confirmed cases of coronavirus at the Institute of Mathematics. As of 5 March, there are 10 confirmed cases at University of Zurich, at least 7 at the I-Math and 1 at the Center of Dental Medicine.[20]"
my own bias is that I find the talk about handwashing etc to be rather like trying to close the barn after the horse has bolted
if you're in a situation where you have to worry about handwashing, then you're probably already in a situation where the likelihood of getting exposed regardless is pretty good
> Undergraduates who live on [MIT] campus must begin packing and moving out of their residences by this Saturday, March 14. This also applies to students in our FSILGs. You will be required to leave by noon on Tuesday, March 17. For first-years, sophomores and juniors: Please pack your belongings and make plans to travel home or to another location off-campus as if you do not expect to return here until the fall semester.
