@LeakyNun you can visualize it by pulling the torus out to an infinite cylinder, then expanding the ends of the cylinder until they come back together. it kind of turns the torus inside out, and ends up swapping inside and outside.
@LeakyNun are you sure you mean GL(n,R) and not GL(n,Z)?
according to Derek Holt, the normal subgroups of GL(n,F) are all subgroups of Z(GL(n,F)) and all subgroups containing SL(n,F), except in the case of GL(2,2) and GL(2,3)
point at infinity, and yes a loop would be null-homotopic. it may help to use R^2 as a model for S^2 as a visual aid. (also you're talking about S^3, not RP^3, methinks/mehopes)
there's a schubert cell decomposition $\mathbb{RP}^n = \mathbb{R}^n \sqcup \mathbb{R}^{n-1}\sqcup\cdots\sqcup\mathbb{R}^1\sqcup\mathbb{R}^0$, but they are not the same as topological spaces if one treats the disjoint union as an operation on topological spaces
I am figuring to hold a panel like discussion involving some regulars of the chat. However a couple of challenges have to be overcame first:
1. I'd like to have a topic on the history of mathematics, but as far I know, no regulars are particularly well versed on that field, and I am not very qualified either as the host because my knowledge on maths history is haphazard
2. Even if we managed to find some, how should it be done, and what is the best date
(Of course I can easily read up a bit to start the topic, but the topic will die if there are not enough people that contribute relevant information. Hmm, I guess I need to think about this more...)
Consider for one car owner the insurance policy with the following clauses:
Deductible: If the loss $X>d$, then the insurer pays only for loss above $d>0$.
Coverage Limit: If the loss $X>l$, then the insurer pays only for loss below $l>d$.
Distorted Distribution: The insurer may base the premiu...
@TedShifrin Hi , got a question for when you're back: Remember the function we talked about the other day $f(w) = \int_{\gamma} 1/(z-w) dz$ ? we saw that $f'(w) =\int_{\gamma} 1/(z-w) \ ^ 2dz $ and we saw that $g(z) = 1/(z-w) \ ^ 2 $ has a primitive , $G(z) = -1 / (z-w)$ so we wanted to conclude that $\int_{\gamma} g(z) dz = 0$ , the problem with it that i see is that $G$ is not a primitive on the ball $B(z_0 , R) $ since $G(w)$ is not defined and cant be continuously defined. how to overcome this?
@NV-US so thinking about this in general, if you have a function $f:\mathbb{R}^n\to\mathbb{R}^m$ (also for an open subset but don't worry about that now), we have that $f'$ should be a function that tells you what the derivative of $f$ is at a point $x$
Formally, this means $f':\mathbb{R}^n\to \mathcal{L}(\mathbb{R}^n,\mathbb{R}^n)$
@KasmirKhaan I can link you the notes I wrote to teach from this semester, though they are structured slightly unusually since I needed to stay within the topics covered previously and had a particular goal in mind
@NV-US just unpack that statement. $f'$ should take a point $x$ in the domain of $f$ and send it to the derivative of $f$ at $x$. So it takes a point and spits out a linear map from $\mathbb{R}^n$ to $\mathbb{R}^n$
Normally I'd hint you through this, but it's short, and if I remember right you're a graduate student. If this is the case, you really want to learn how to execute these sorts of linear algebra arguments independently before it really bites you
@PyRulez As I see the sentence "Category theory is cool." in your profile, maybe it's worth mentioning that category theory chatroom has been created recently. Let's hope some users interested in the topic start visiting this room and the activity in the room will increase.
I can say that T(x) = Ax, where A is a 1x1 matrix (a scalar), from the fact that if T: R^n -> R^m then T(x) = Ax where A is a mxn matrix. I can prove this fact. Will this do? @Daminark
That is valid, though it still is good practice to prove this straight from the definition. If you know the proof of the latter fact already, it specializes immediately
If not, it still can be done (and you should try to tackle the general fact as well)
@NV-US if you are feeling like your linear algebra needs some brushing up, I would recommend finding a book on that and reading it before trying to continue in multivariable calculus
Perfect. Now, let's specialize to the case we were talking about earlier, if $T:\mathbb{R}\to\mathbb{R}$
We want to find $T(a)$, and we know that for any $x$ and $c$, we have that $T(cx) = cT(x)$. What would be a natural choice for $x$ and $c$ here that could help us?
Now let's return to the analysis situation. I want you to take me on faith for now that $f':\mathbb{R}^n\to \mathcal{L}(\mathbb{R}^n,\mathbb{R}^m)$, and let's plug in $m=n=1$
So $f':\mathbb{R}\to \mathcal{L}(\mathbb{R},\mathbb{R})$
But remember that a linear transformation from $\mathbb{R}$ to $\mathbb{R}$ is defined by a single number, namely $T(1)$, as you just proved
So $\mathcal{L}(\mathbb{R},\mathbb{R})$ is in fact isomorphic to $\mathbb{R}$
That's why you can think of it in this way
"it in this way" meaning that $f':\mathbb{R}\to\mathbb{R}$
This is why I said that you seem to be making the wrong generalization from single variable to multi, since if you look at that, you may read it off as "okay $f'$ has the same domain/target space that $f$ does", which isn't the case
Geometrically, you say that at a given point, $f'$ tells you what the slope of the tangent line to the curve at $x_0$ is. The slope of the tangent line determines the tangent line because of your argument above
if $f: R^n -> R^M$, so does it not follow that $f' : R^n -> R^m$, it follows in single variable, but not in multi variable? this is what u meant to say? do u have any example?
Yes, that's true always. The reason is because of what I said some time back: the derivative should take in a point and spit out the derivative of $f$ at that point, which is a linear map
It's just a freak accident that if the dimension of everything is $1$, the two correspond, but in general it's not the case, and literally any derivative in higher dimensions will be your example
So the idea behind trying to find it comes from intuition
The derivative is a linear approximation to a function near a point. So a linear approximation of a linear function should be linear
This is one of those proofs where you want to reason for a bit and try to find a suspect for the derivative
Once you have a suspect, try it to see if it works. In this case it was relatively simple, and it was stated in the question, but in general it may not work, in which case you try to find what failed, look back at your reasoning, etc
It's another general principle that I'd recommend adopting, at least at this stage. If you're trying to prove an existence statement, see if a natural suspect just jumps out at you, either immediately or after having reasoned about the problem for a while
Right now nothing is rigorous, my explanation of the above was a heuristic that one would use to inspire the guess that the derivative of a linear function is linear, not a rigorous proof (which is what you did)
Well, $f$ obviously approximates $f$ in that case, since $|f(x)-f(x)| = 0$
Also, one thing I never got to respond to earlier was when you said that eventually you'll get a linear algebra book but that you have exams in the short run
I realize the temptation to engage crisis-management mode, and be focusing on the most urgent matter at hand, but I'd strongly encourage breaking out of it
I'm not sure if you'll be able to handle exams in multivariable calculus without a strong working knowledge of linear algebra. If you don't have the time to get both up to par, then you may be screwed one way or another, I'd consult your professor
The level of mastery required may vary from place to place
But on general principle, people should have a good understanding of linear algebra and be able to work with it easily before trying to approach calculus, since the latter seriously relies on the former
I don't know the details of your situation and how your exams will look like, maybe there is a way to get around it temporarily
But, and I do not say this with any intention to offend, the fact that you had some difficulty with the linear algebra question I asked above is somewhat worrying, in multivariable calculus, especially in a grad class, they won't likely forgive this lack of understanding
Ah, that's a bit of a problem. I'd recommend consulting your multi teacher as soon as possible to let him/her know of the situation, and take whatever advice you're given to heart
I used Hoffman and Kunze and liked it for the most part, but it can be a bit tricky since it requires being approached with a certain mindset that isn't shared by all
I don't usually recommend this book, but it may be helpful in your case: Look up linear algebra done right
Artin is good for an integrated treatment of the two, so you wouldn't need to get two different books for groups/rings/fields and standard-fare linear algebra
Translation: A pond has $500000$ liters of water in it. The pond contains $y$ liters of poison after $t$ years. Assume that the poison is evenly distributed throughout the pond. The water flows out of the pond with the speed of one tenth per year of the actual amount of water, and new water ...
Consider for one car owner the insurance policy with the following clauses:
Deductible: If the loss $X>d$, then the insurer pays only for loss above $d>0$.
Coverage Limit: If the loss $X>l$, then the insurer pays only for loss below $l>d$.
Distorted Distribution: The insurer may base the premiu...
Translation: A pond has $500000$ liters of water in it. The pond contains $y$ liters of poison after $t$ years. Assume that the poison is evenly distributed throughout the pond. The water flows out of the pond with the speed of one tenth per year of the actual amount of water, and new water ...
