I mean for example: given two particles of mass m and M which are distance r apart, the gravitational force between them is given by Newton’s law of gravitation as: $\frac{GmM}{r^2}$.
We consider a star of mass M and a planet of mass m. If the planet revolves around the start then assuming circular orbit, the planet needs a centripetal force.
Now the question is: Has this ever been demonstrated on a lab scale? Because on lab scale, masses will be extremely small and hence feeble gravitational force. So I think that it can’t be demonstrated.
(not that all of astronomy is applied gravitational mechanics: for instance, using spectral lines to deduce the composition of stars. but i'd still say that's the majority of it)
Probably not much is known about gravity yet. Newton’s universal law of gravitation does say that the force has so and so value but does not explain why the force exists.
there have been questions about whether gravity gets modified at small distances. people who wanted extra dimensions to be a thing were hopeful about that, but...nah
it seems like it would be hard to get one thing orbiting around another thing in an actual lab. the forces involved would be so tiny and any friction would dampen the festivities.
i will say: the fact that Newtonian gravitation works well (but not perfectly!) up to the planetary level, and apparently works down to the width of human hair
I want to ask how this makes sense: finding largest positive a such that: $\int_0^3 f + \int _0^7 f^{-1}\ge a$ for every continuous and surjective $f:[0,\infty)\to [0,\infty)$.
finding largest a? What does that mean?
Suppose it is 21, then I can find some f such that LHS is 25.
it is also given that f is increasing in the strict sense.
@robjohn Is there a way to explain why changing the order of integration in this answer appears to be valid? I could integrate by parts before changing the order of integration, but then things just get messy.
@RandomVariable The only thing I can see would be to integrate by parts, change order, and hope that integrate by parts will simplify things after. That looks like a touchy integral.
landon: it's kinda fun to see how high n can get before wolfram alpha or your favorite software no longer evaluates int 1/(x^n + 1) dx (or its variants, e.g. with your +2) exactly.
last i checked it didn't occur to WA to try series as an alternative.
But I am far from being an expert myself. I just need calculus on occasions. This one seemed quite funny because for the uninitiated the result mihgt look very surprising
@VLC Consider the following transformation of the plane: $(x, y)\mapsto (y, x)$. Now introduce the usual polar coordinates after this tranformation. This is basically what you are doing.
sometimes problems like this have cute generalizations if the inverse multifunction is not too badly behaved. i dunno about this one, though. my mental picture really needs it to be the way it is.
i used to work with an economist who had a book full of multifunction-based inequalities. economists need them for some reason. i guess because what goes up must come down.
unless it's property values. those only go up, right?
i wish i knew what that book was called. it was a type of book i hadn't seen before. kinda a long list of formulas and theorems and identities, with a lot of emphasis on explicit equations for everything. but some proofs and pretty broad coverage. game theory one page, schauder's fixed point theorem on the next.
@robjohn I basically asked about that a few months ago here. I don't really understand the self-contained proof in the accepted answer, but the "quick proof" is what I was looking for.
under: that's a good strategy. i used to do that, at least if i had a half-formed strategy with an identifiable gap in it that was not equivalent to the original problem.
i liked seeing that when grading, too. it conveys more information than a skipped problem.
Yeah, I try not to do that. I did the proof up until where I didn't where to go. Then I described what I thought the next step might be, but explained I wasn't certain.
i think sometimes the better the student, the more they feel like they have to be perfect or skip the problem. struggling students are more than happy to fill a page.
