dy/dw is a one variable derivative of the function f(g(w),w). defined as a one variable limit of difference quotients. partialy/partial x would more preferably be written partial f/partial x and that is a partial derivative, limit of difference quotients in the first coordinate of the two variable function f, leaving the second fixed. same with partial f/partial y and the second coordinate.
unless your book has given independent meaning to differentials, which it may not have at this point in the treatment (?), no need to interpret partial y or dy on their own.
I have a question in the proof of Thm 9.5. In the middle of the proof, it shows that the ring of integers (integral closure of $\Bbb Z$ in $K$ where $K/\Bbb Q$ is a finite field extension). Then it says that $A$ is integrally closed by (5.5)
(5.5) : Let $A\subset B$ be rings and let $C$ be the i...
in my more favored notation i would create a separate name for the one variable composite function, something like h(w) = f(g(w),w), and write the thing to be proved as h'(w) = f_1(g(w),w) g'(w) + f_2 (g(w), w). no differentials anywhere, just derivatives, of a single variable function g and partial derivatives of a two variable function f.
I don't understand in the proof of 'the ring of integers in an algebraic number field is dedekind', in particular showing the ring of integers is integrally closed, the corollary (5.5) come out
if you're being asked about differentials in their exams, you'll be told what they are before then. a lot of real analysis books don't introduce them, but they do introduce $\partial$ and $d$ notation, which can be confusing if you know that differentials can be given formal meaning but haven't been told how.
leibniz notation does not mandate suppressing the points at which you are evaluating partial derivatives, but it seems to encourage people to do it. note that the more explicit way of writing it makes clear what is being held constant when you take the partial derivatives of f.
i don't know that the problem statement implies that there is a single integer m for which f takes on each value at most m times. if it somehow does that seems like a separate argument.
@TedShifrin Oh my God!! Really ?? I never noticed. I'll take a look right now. Sir, I have heard about your book though and I'm also aware that your video lectures are available online.
@leslietownes: So except that $m$ part, I take everything else is correct. In fact, we may ignore $m$ altogether as I never used it in my proof. Right?
@BalarkaSen I have heard that you cleared the TIFR GS i-phd entrance exam in '21. First of all, many Congratulations for that! So, can you please share some questions from Algebra and Analysis which they asked in your interview in TIFR, Mumbai!
my daughter got to ride on a pony this morning. a few hours later, the only thing she remembers is that the pony urinated at one point. she can't stop talking about it.
i used to bring my kids to oakland zoo more or less once a week. one week the giant tortoises were in heat and hard at it, along with unmistakable sounds that could be heard around the entire zoo. so we just had to go to observe the slow motion antics.
I have updated the usage instructions for ChatJax on the installation page so that it will hopefully be easier for Chrome on Android. Dima Pasechnik on MathOverflow helped greatly and Use Bookmarklets on Chrome on Android was also very helpful.
@Koro: Well, you never drew the picture, but you had the right idea. I don't understand why the proof is so complicated. By the intermediate value property, the function has to be within the $\epsilon$-band if $|x-a|<\delta$. I admit I did not read your argument.
@TedShifrin: I did draw the picture though not uploaded it and that's how I got the idea actually. The idea being that there's a strip around $a$ where $f$ is either strictly increasing or decreasing.
I don't need that to get the explicit $\delta$. I get the same $\delta$ that you have. And, yes, I need a little contradiction argument to show that $|f(x)-f(a)|<\epsilon$ for $|x-a|<\delta$.
so, I have $g(x) = 2x$ and $f[g(x), x] = g(x) + 3x$, so I have $\Delta f = g'(x) \Delta x + 3 \Delta x + \epsilon \Delta x$, where $\epsilon \Delta x$ is the error representing the difference between $df$ and $\Delta f$. Does $\Delta f$ have some known name, i.e. is it any important function in analysis?
when my daughter was young she was fascinated by horses, so we sat down and typed in horses and searched. was not expecting a graphic depiction of catherine the great to pop up.
@robjohn yes, that is exactly how I think of cos and sin (literally everytime I need to evaluate a trig function I draw a unit circle and then throw all hard work out the window by rounding to the nearest $\frac{\pi}{4}$)
fixing x, and some increment h, Delta f is just f(x + h) - f(x)
it doesn't have a nicer name. it can certainly be written in terms of other Deltas plus an error, but that's just what it is. unless i'm missing something.
i really think it's helpful not to use notation that suppresses where things are being evaluated and what the increment is. using "Delta x" as a name for a quantity that can vary independently of x (i call it h) is traditional but in my opinion a bad idea.
some people become very skilled in properly leibniz notation, deltas, ds, etc. without ever worrying about this stuff or even noticing it. i am not among those people, i need to learn what my notation is forgetting before i forget it.
hello. suppose I have a sum $S(N):=\sum_{n=1}^{N}f(n)$, which isn't convergent (oscillating infinitely many times) but it takes a finite value $c$ infinitely many often. is there some kind of "limit" where we can say $L=\lim_{N\to\infty}S(N)$ so that $L= c$? (not talking about liminf/limsup) Is this "limit" meaningless? Does it even make sense?
Sometimes I'm very tempted to write this: If for every $\epsilon\lt 0$ there exists a $\delta\gt 0$ such that $|x-a|\lt \delta\implies |f(x)-f(a)|\lt -\epsilon$ then $f$ is said to be continuous at $a$ and we write $\lim_{x\to a}f(x)=f(a)$.
note that there might be infinitely many c for which the partial sums assume the value c infinitely often. in such an instance your 'limit' would have to pick one. nothing wrong with doing so, but it gets subtle if you want your "limit" to behave like a limit (i.e., have similar algebraic properties) on a set of sequences that is larger than the single one you are considering.
the shift invariance of banach limits sometimes limits the possible values that the banach limit can take on a non-convergent sequence. given S, you might not be able to find a banach limit L with L(S) = sup {S_n: n in N} for example, even if the sup is attained infinitely many times.
for example if you want to extend "lim" from convergent sequences in a shift-invariant and linear way to a set of sequences that includes (-1)^n, your "lim" must assign the sequence (-1)^n the value 0. you can't give it the value 1 (or -1) without breaking shift invariance or linearity
Specifically, "[in ref to a generic triangle $ABC$] Let $p,q$ be the radii of two circles through $A$, touching $BC$ at $B$ and $C$, respectively. Then $pq = R^2$ [where $R$ is the circumradius]."
@leslietownes Interesting. What if I have an error term $\espsilon(N)$, which would go to zero as $N\to\infty$. (In this case I'm talking about a "normal or usual limit"). Furthermore $F(N):=S(N)+\espsilon(N)$, where $S(N)$ denotes my previous defined sum and it takes the values $c_1$ and $c_2$ infinitely many often ($c_1\neq c_2$). Now let the "special limit" go to infinity and we'll arrive to $F(N)=c_1=c_2$, a contradiction? How should I handle this? Does the Banach limit consider such a case?
i don't see how you compute with your 'special limit' without assuming more about it than i have seen. maybe i have missed something.
a banach limit would assign the same value to F and S but at this level of generality (assuming only that S assumes some value infinitely many times) you would not be able to get your hands on what that value would be
fargle as i read it, you've got a circle that goes through both A and B, and a circle that goes through both A and C. they have some radii p, q - which are not determined by this information, but somehow their product is, and is the square of the circumradius of the triangle.
Consider the optimization problem of maximizing the Sharpe Ratio, given a riskless asset with the return $r>0$ and $n = 5$ possible risky assets whose expected return is $\mu$ and covariance matrix of returns is $Q$. Assume that the amount invested in the third asset should be no greater than the...
@AndrewMicallef @leslietownes, so I didn't get very far last night on this. I'm focusing on playing with the lefthandside (ie the trig functions) but not sure what I can do to get $it$ out of $\cost + i\sin\t$.
Ah fair enough, maybe I will come back to this after taylor, it just poped into my head after I was thinking about what happens when you just use silly approximations
ted, last night my stab at a geometric way was to do cos and sin geometrically and then just define e^(it) via that formula. so you can deduce various exponential identities from everybody's favorite pictures of triangles.
this obviously was not well adapted to proving the identity, and of no help at all. but defining one's way out of a box is always an option.
Crap. I just tried the mobile site and I don't see the sidebar at all. I can get site info, but the links for the guidelines and LaTeX in chat are not clickable. How do people get the bookmark on the mobile site these days? I can get the full site, but that is crappy that we can't see the sidebar info.
not a huge fan of straight white rice. until recently a friend made me a pound of fresh pasta every week, was the culinary highlight for me, just with basil pesto for the most part.
I have the following differential: $df = \lim \limits_{h \to 0} f[g(w+h)] - f[g(w)]$ after staring at this equation for about 40 minutes I have had the flash of insight that this could be$\frac{\partial f}{\partial g} \frac{\partial g}{\partial w}$, but I have no clue why. it just feels like for a small enough $h$, somehow multiplication would magically work like function composition. am I completely wrong?
Do linear approximation of $g$ inside $f(g(w+h))$. A proof of the chain rule is in every decent analysis book (including my own or my YouTube lectures in the multivariable case).
