Given how many people ask questions in here and on the site when they don't even have any idea what the definitions of the terms in their questions are ...
i'm very pleased at a very minor accomplishment. i've been trying to answer a question all day in a different way to the current answer and was stumped until a few minutes ago.
google sometimes now generates 2d plots if you ask it to. i mentioned this the other day. i'm still blown away by it. i.e. who is the audience, who doesn't already have other better stuff.
depends on how you define cos(t), sin(t), and e^(it) for real t
you can find a lot of proofs out there, but if they use different definitions than whatever you've been asked to use, they may not count as proofs
to give an example of what i'm talking about, sometimes cos(t) is defined in terms of a ratio of sides of a triangle inscribed in the unit circle. sometimes it is defined as the inverse of a definite integral of 1/sqrt(1-x^2) dx. sometimes it is defined in terms of a power series.
how it's defined affects how you prove things about it
i've seen books that develop sin and cos in elementary fashion, and then use that equation as the definition of e^(it) for real t. in those books there is nothing to prove, it's just what e^(it) means
huh, I had no idea sin and cos had different definitions, and on reflection had no idea what the definition of cos was.
I've always thought of them in terms of the unit circle
well the lecture I was just watching briefly brought up the question I just posed, in terms of a unit circle, so I supose that is the method I should use.
the unit circle definition has simplicity on its side but it's not very congenial to proving things with. you have to draw triangles and stuff. it also doesn't readily lend itself to numerical approximation. outside of 'special angles' where exact values follow from special triangles. how would you even get a few digits of cos(1) for example.
the integral definition of cosine is also amenable to numerical approximation, you don't necessarily need the series. i don't have a favorite definition, they all have plusses and minuses
it wouldn't surprise me if they just define it using that formula. the addition laws for the exponential function then follows from the addition laws for trig functions, which do have relatively OK picture proofs. you do have to think a bit to see why a picture proof for angles in some quadrant would work the same way for any real angles and not just ones lying in that quadrant. but it works well enough.
people do like the taylor series definitions of everything because they make these kinds of calculations fairly routine, at least when the functions' taylor series are simple enough, as they are here. but proving the basic facts you need to manipulate series of complex numbers is maybe harder than just using that identity as a definitional starting point.
have fun
one of the more amusing pairs of definitions of sin and cos is that they are the solutions to y'' + y = 0 satisfying y(0) = 0, y'(0) = 1 [sin] and y(0) = 1, y'(0) = 0 [cos], respectively. you need some ODE theory to make sense of that but you can show that you do indeed get the unit circle thing from that.
there was a pretty good expository article i read once that ran through a bunch of definitions of these things and proved them equivalent.
Maybe they are some people around here who use MathJax in chat on their mobile phones and would be able to give some advice to this user in the MathOverflow chatroom:
yes, this appears to work at the moment in Firefox 87.0 on Linux (it used to be broken). But I need to click on this bookmark every time I visit this page. I fear it won't work on my Android phone (pretty important for me).
Just tried creating such a bookmark on Android (Mobile Chrome browser). While I can create a bookmark with the javascript code as proposed, it does not function at all.
I imagine one should create a proper Chrome browser plugin to support this in such a setting.
Let $\vec{x} ,\vec{y},\vec{z}$ be three vectors 9f magnitude $\sqrt{2}$ and angle between each pair of them is $\pi/3$. If a is a non zero vector perpendicular to $\vec{x} $ and $\vec{y}×\vec{z}$ and $\vec{b}$ is a non zero vector perpendicular to $\vec{y}$ and $\vec{z}×\vec{x}$, then..?
that's not what they seem to be talking about, though when they say "constant is 1". i say this because that equation with I_n has been used in several places as an equation (never an inequality), and it was first seen in fritz john's theorem
Is this site designed to ask questions that touch on hyper-specialized hot research that may confuse neophytes or should such questions be reserved for specialists in the subject matter outside of this site? (My problem is that I work alone.)
The problem is obviously for me, not for the site if I understand it better (I just arrived on Maths Stack Exchange)
@robjohn :ok. I use a numerical system that is not well known (there are some results published by the OEIS for example). The name of this system is Primorial number system or primoradic. I could see that some users of the site had some knowledge about this system and so I asked a more difficult question which I think was not well understood.
Consider | x + 5 || x - 4 | < ε, where ε is infinitesimal then if | x + 5 | < β then how is it possible to write | x - 4 | < ε / β, i am unable to digest. Help!
@robjohn : That is what i did. Links towards stub OEIS, towards other questions I asked where i expose more details about primoradic...
@robjohn link towards primorial expansion of e
@robjohn : The question was then clearly only addressed to people who mastered primoradic and I am afraid that it attracted the wrath of people who were confused by the use of such a numerical system for issues that are otherwise of interest to them. I got a flood of down votes without the slightest request for clarification that I would have gladly given.
@robjohn : i already developped the Wikipedia page in French dedicated solely to primoradic (and quite extensively) and gave a link in my previous questions.
@robjohn : for me, there is the evidence of the drawing. It has just to be formalized a bit.
@robjohn : I (re-)discovered by myself primoradic. I created it by myself. I didn't know that it exists already. I developed it alone. In the context of research that I have been conducting alone for several years.
@robjohn : It is a digital system that has become extremely natural to me.
@epsilon-emperor "be so sure"? I mentioned that "inequalities are often in the form..." in response to your query "How did you figure this out? What's the context".
The paper says "the correct constant in the inequality is 1 (as written)." and there is only one inequality
Let $S$ be an open bounded set, then the set $T(S)$ where $T$ is the family of affine transformations is a basis for the euclidean topology right? we are working in $\mathbb R^2$
I'm currently reading the proof of the following fact
Let $K/F$ be a field extension of finite degree $n$. If $K/F$ is separable, then the trace form $T:K\times K\to F$ is nondegenerate. Here, trace form is defined by $T:K\times K\to F$ by $T(x,y) = \operatorname{Tr}(xy)$
Proof. By the primitiv...
@Thorgott You mean $\sigma_1(\alpha),...,\simga_n(\alpha)$ are conjugates of $\alpha$ then $\sigma_1(\alpha^N),...,\sigma_n(\alpha^N)$ are conjugates of $\alpha^N$?
but if $L/K$ is a separable extension and $\sigma_1,\dots,\sigma_n$ are the $n$ distinct embeddings of $L$ into a separable closure of $K$, then $\mathrm{tr}(\alpha)=\sum_{i=1}^n\sigma_i(\alpha)$ for any $\alpha\in L$
@Thorgott Yes what I meant is that I understood $\sigma_1(\alpha^N),...,\sigma_n(\alpha^N)$ are conjugates of $\alpha^N$ but need not distinct and I thought it's problematic because I didn't know that in the definition of $Tr$, they count multiplicity.
I have a question concerning multiple integrals. How am I supposed to interpret the output of an indefinite multiple integral given its domain and some input value from the domain?
I understand that it is the plane of the domain and the surface defined by a function, but what is, for example, in the indefinite integral of some function $f(x,y)$, the evaluation of $f(x,y)$ for some $x$ and some $y$ referring to what region under the surface?
@TedShifrin, what would $x \cdot \nabla f$ look like in polar coordinates? $x \cdot \nabla f = x_1 \partial_1 f + x_2 \partial_2 f$ in cartesian coordinates, but how does it look in polar coordinates? $\cos (\theta) \partial_r f + \frac{1}{r} \sin (\theta) \partial_{\theta} f$?
Let $\vec{x} ,\vec{y},\vec{z}$ be three vectors 9f magnitude $\sqrt{2}$ and angle between each pair of them is $\pi/3$. If a is a non zero vector perpendicular to $\vec{x} $ and $\vec{y}×\vec{z}$ and $\vec{b}$ is a non zero vector perpendicular to $\vec{y}$ and $\vec{z}×\vec{x}$, then..?
@JoeShmo You need to do a chain rule calculation to figure out, say, gradient in different coordinate systems. I prefer to do it with differential forms. If $e_r,e_\theta$ are the unit vectors in the radial and tangential directions, then the dual $1$-forms are $dr$ and $r\,d\theta$. You write $df = f_r\,dr + f_\theta\,d\theta$, so the gradient is the vector field $f_r e_r + (f_\theta/r) e_\theta$. Etc.
I have a function that satisfies intermediate value property and in addition has the property that it takes each value exactly once
then that function is to be proven to be continuous.
Here's the proof outline that I followed: 1) First I proved that this function (henceforth called $f$) is monotonic in strict sense, whence it follows that one-sided limits (L and R) exist at every point. 2) WLOG, I assumed strictly increasing and we know the relation $L\le R$ for strictly increasing functions. On the contrary I assumed $L\lt R$ and got a contradiction thereby proving $L=R$
I'm having a confusion over the veracity of the statement that a function that satisfies the Intermediate Value Theorem and takes each value only finitely many times is continuous.
I've seen from a problem in Spivak's Calculus that this statement is true.
Proof: For the case where $f$ takes o...
my idea wasn't as short as the solution above, but it did involve monotone sequences in the domain approaching a point where a one-sided limit didn't exist.
@TedShifrin Infact, that's why I asked this question here. I was also looking for a method to solve it without contradiction. Because I strongly believed contradiction can solve it.
This is because the first part of it was: $f$ takes on each value exactly once and follows IVT
I'm having a confusion over the veracity of the statement that a function that satisfies the Intermediate Value Theorem and takes each value only finitely many times is continuous.
I've seen from a problem in Spivak's Calculus that this statement is true.
Proof: For the case where $f$ takes o...
