quotient Definition and 341 Threads

A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of...

If I'm not mistaken, the curl can be expressed like this in index notation: ##(\nabla \times \vec v)^i = \epsilon^{i j k} \nabla_j v_k = \epsilon^{i j k} (\partial_j v_k - \Gamma^m_{j k} v_m) = \epsilon^{i j k} \partial_j v_k ## (where the last equality is because ##\epsilon^{i j...

Hey all, I have a very simple question regarding the quotient of complex values. Consider the function: $$f(a) = \sqrt{\frac{a-1i}{a+1i}}$$ where ##i## is the imaginary unit. When I evaluate f(0) in Mathematica, I get ##f(0) = 1i##, as expected. But if I evaluate at a very small value of ##a##...

Please, in the definition of quotient Lie algebra If ##I## is an ideal of ##\mathfrak{g}##, then the vector space ##\mathfrak{g} / I## with the bracket defined by: $$[x+I, y+I]=[x, y]+I, for all x, y \in \mathfrak{g}$$, is a Lie algebra called the quotient Lie algebra of ##\mathfrak{g}## by...

Closer to odd number implies ##|y/x - (2n+1)| < 1/2## for ##n = 0,1,2...##. Then $$-\frac 1 2 < \frac y x - (2n+1) < \frac 1 2 \implies\\ y < (2n + 1.5)x,\\ y > (2n + 0.5)x$$ for each ##n##. We note ##x \in (0,1)## implies ##y## can be larger than 1 since the slope is greater than 1 (but we know...

I just spent 15 minutes to re-type all the Latex again because I lost everything while editing. why does this happen?? This is a huge waste of time. ##f## is entire so ##f## is holomorphic on ##\mathbb{D}∪C##. Also, ##\mathbb{D}∪C## is a connected set. By the maximum principle, ##f## restricted...

Hello! I have two related exercises I need help with 1. Partition the space ##\mathbb{R}## into the interval ##[a,b]##, and singletons disjoint from this interval. The associated equivalence ##\sim## is defined by ##x\sim y## if and only if either##x=y## or ##x,y\in[a,b]##. Then...

Hello! Reading a textbook I found that authors use the same trick to show that subsets of quotient topology are open. And I don't understand why this trick is valid. Below I provide there example for manifold (Mobius strip) where this trick was used Quote from "Differential Geometry and...

Hi Guys Sorry for the rudimentary post. I am busy with a numerical solution to a mechanics problem, and the results are just not as expected. I am re-checking the mathematics to ensure that all is in order in doing so I am second guessing a few things Referring to the attached scan, is the...

Let ##n=\dim X## and ##m=\dim Y##. Define a basis for ##X: y_1,...,y_m,z_{m+1},...,z_n##. The first ##m## terms are a basis for ##Y##. The remaining ##n-m## terms are a basis for its complement w.r.t ##X##. Let's call it ##Z##. ##X## is the direct sum of ##Y## and ##Z##; denote it as ##X=Y+Z##...

%%% Assume that ##X/Y## is defined. Since ##\dim Y = \dim X##, it follows that ##\dim {X/Y}=0## and that ##X/Y=\{0\}##. Suppose that ##Y## is a proper subspace of ##X##. Then there is an ##x\in X## such that ##x\notin Y##. Let us consider the equivalence class: ##\{x\}_Y=\{x_0\in...

I see that the first four equations are definitions. The problem is about the dimensions of the quotient. Why does the set Kx forms a six dimensional Lie algebra?

Good Morning I have read that it is not justified to split the "numerator" and "denominator" in the symbol for, say, dx/dt However, when I look at Wikipedia's discussion on the Principle of Virtual Work, they do just that. (See picture, below). I was told it is OK in 1D cases, but note the...

Hi! I want to check if i have understood concepts regarding the quotient U/V correctly or not. I have read definitions that ##V/U = \{v + U : v ∈ V\}## . U is a subspace of V. But v + U is also defined as the set ##\{v + u : u ∈ U\}##. So V/U is a set of sets is this the correct understanding...

Let V = C[x] be the vector space of all polynomials in x with complex coefficients and let ##W = \{p(x) ∈ V: p (1) = p (−1) = 0\}##. Determine a basis for V/W The solution of this problem that i found did the following: Why do they choose the basis to be {1+W, x + W} at the end? I mean since...

hello Witch of these are certain sentences? a-\dfrac{e}{m_e}>\dfrac{H^{-}}{m_{H^{-}}}\cdot{1000} b-\dfrac{e}{m_e}>\dfrac{H^{+}}{m_{H^{+}}}\cdot{1000} The first accurate measurement of e/m was made by english physicist J.J. Thomson in 1897, who demostrated that the quotient charge-mass of the...

Not really a homework problem, just an equation from my textbook that I do not understand. I can't think of any way to even begin manipulating the right hand side to make it equal the left hand side. Just to confirm equality (thanks to another user for suggestion), I multiplied both sides by of...

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The following thread regards how I am to receive an accurate gauge of my IQ. Heretofore, I undertook several IQ tests, namely the Serebriakoff Advanced Culture Fair Test, Numerus Basic, Mensa Norway, Mensa Denmark, Tero 41 and Logica Stella. Additionally, I took the Mensa Luxembourg Online Test...

Now, I understand how to use the quotient rule for derivatives and everything. I do not struggle with using it, my question is mostly about the formula itself...I very much enjoy WHY we do things in math, not just “here’s the formula, do it”...Here is the formula for the quotient rule of...

By comparing the given equation to the equation for the Sturm-Liouville form, I see that p(x) must equal 1, q(x) must equal 0 and sigma(x) must equal 1. After this, I have no idea what I should be doing for part 2.

Homework Statement lim (1/x - 1/3) / (x-3) x->3 Homework EquationsThe Attempt at a Solution I tried to cancel the bottom (x-3) out by multiplying the top by 3/3 and x/x and then got ((3-x)/3x)/(x-3) but ended with 0/0 and the right answer is -1/9. The top part is confusing me.

Homework Statement For a commutative ring ##R## with ##1\neq 0## and a nonzerodivisor ##r \in R##, let ##S## be the set ##S=\{r^n\mid n\in \mathbb{Z}, n\geq 0\}## and denote ##S^{-1}R=R\left[\frac{1}{r}\right]##. Prove that there is a ring isomorphism $$R\left[\frac{1}{r}\right]\cong...

Mentor note: Thread moved from technical math section, so is missing the homework template .

Homework Statement What is the step by step difference quotient instructions of d/dx [ln (x+3)]? Homework EquationsThe Attempt at a Solution I tried to solve but i got as far as (lim h--> 0 ..of.. 1/h * ln (x+h+3/x+3)[/B]

Hi PF! Given the ODE $$f'' = -\lambda f : f(0)=f(1)=0$$ we know ##f_n = \sin (n\pi x), \lambda_n = (n\pi)^2##. Estimating eigenvalues via Rayleigh quotient implies $$\lambda_n \leq R_n \equiv -\frac{(\phi''_n,\phi_n)}{(\phi_n,\phi_n)}$$ where ##\phi_n## are the trial functions. Does the...

Suppose that we know that ##G=\langle S \mid R\rangle##, that is, ##G## has a presentation. If ##N\trianglelefteq G##, what can be said about ##G/N##? I know that for example, if ##G=\langle x,y \rangle##, then ##G/N = \langle xN, yN \rangle##. But is there anything that can be said about the...

Homework Statement Find out the quotient derivative i.e. the derivative of polynomial upon polynomial and then find the minima and maxima.[/B] ##W\left(z\right)=\frac{{4z+9}}{{2-z}}## Homework Equations ##\left( \frac{f}{g} \right)' = \frac{f'\,g - f\,g'}{g^2}## The Attempt at a Solution...

I am newbie to topology and trying to understand covering maps and quotient maps. At first sight it seems the two are closely related. For example SO(3) is double covered by SU(2) and is also the quotient SU(2)/ℤ2 so the 2 maps appear to be equivalent. Likewise, for ℝ and S1. However, I...

So I'm just beginning to study abstract algebra and I'm not sure I grasp the definition of a quotient group, I believe it probably has to do with the book providing little to no examples. In trying to come up with my own examples, I imagined the following: Consider the Klein four group, if we...

Let ##M## be a (right) R-module, and ##A## and ##B## two submodules of ##M##. If ##A = B ##, then we know that ##\frac{M}{A} = \frac{M}{B}##. But is the converse also true: If ##\frac{M}{A} = \frac{M}{B}##, can we conclude that ##A = B ## ? I doubt it, but I cannot find the answer. Maybe...

Homework Statement "Suppose ##f:(a,b) \rightarrow ℝ## is differentiable at ##x\in (a,b)##. Prove that ##lim_{h \rightarrow 0}\frac{f(x+h)-f(x-h)}{2h}## exists and equals ##f'(x)##. Give an example of a function where this limit exists, but the function is not differentiable." Homework...

I am reading "Algebra: An Approach via Module Theory" by William A. Adkins and Steven H. Weintraub ... I am currently focused on Chapter 2: Rings ... I need help with fully understanding some remarks by Adkins and Weintraub on quotient rings on page 59 in Chapter 2 ... The remarks by Adkins...

I am reading "Algebra: An Approach via Module Theory" by William A. Adkins and Steven H. Weintraub ... I am currently focused on Chapter 2: Rings ... I need help with fully understanding some remarks by Adkins and Weintraub on quotient rings on page 59 in Chapter 2 ... The remarks by Adkins...

The quotient limit laws says that the limit of a quotient is equal to the quotient of the limits. If we had a limit as x approaches 0 of 2x/x we can find the value of that limit to be 2 by canceling out the x’s. If we split it up we get the limit as x approaches 2 of 2x divided by the limit...

Homework Statement What is the integral of e(x)2 over (x-1)2Homework Equations integral of (x-1)-2 is -(x-1)-1 and derivative of e(x)2 is 2xe(x)2 The Attempt at a Solution I tried to integrate it by part but I couldn't get a solution. I want to know how to start solving this question.

These are the two things that I'm going over in my PreCalculus class- the Difference Quotient and the Average Rate of Change of a Function. I'm confused as to what exactly they are, and how they relate to each other. Average Rate of Change= ##\frac {f \left(b\right) - f\left(a\right)} {b - a...

Homework Statement Give an example of a commutative ring ##R## and ##f(x), g(x) R[x]## with ##f## monic such that the remainder after dividing ##g## by ##f## is not unique; that is, there are ##q,q',r,r' \in R[x]## with ##qf + r = g = q' f + r'## and ##\deg (r)## and ##\deg (r')## are both...

This is not homework, it's self study material. I would rather post it here than where questions are usually posted (homework help section) because i think it's much more likely to be seen here by somebody with knowledge on the subject. Let G be a topological group acting continuously on a...

Say H is a subgroup of topological group G. Let L_g: G--->G be denote a map of G acting on itself by a left translation of g. Show that L_g passes(descends) to the quotient G/H. I am a bit confused here, for L_g to pass to the quotient G/H, it would have to be constant on the fibers of G/H...

I know how to prove the quotient rule by using the definition of a derivative using limits (Newton's style). I just saw a proof of the product rule using Leibniz's concept of differentials on Wikipedia. https://en.wikipedia.org/wiki/Product_rule#Discovery Does anyone know of a Leibniz-style...

Homework Statement Consider the quotient ring ##F = \mathbb{Z}_3 [x] / \langle x^2 + 1 \rangle##. Compute the order of the coset ##(x+1) + \langle x^2 + 1 \rangle## in the group of units ##F*##. Homework EquationsThe Attempt at a Solution I was thinking that I just continually compute powers...

Homework Statement Find the order of ##\mathbb{Z}_3 [x] / \langle x^2 + 2x + 2 \rangle ## and ##\mathbb{Z}_3 [x] / \langle x^2 + x + 2 \rangle ## Homework EquationsThe Attempt at a Solution Is there an efficient method for doing this? Is the answer 27 for both? It would seem that both of these...

Homework Statement Sorry for the multiple postings. I actually solved the other problems, so I have this last one: Are ##\mathbb{Z}_3[x] / \langle x^2 + 2x + 1 \rangle ## and ##\mathbb{Z}_3[x] / \langle x^2 + x + 2 \rangle## isomorphic Homework EquationsThe Attempt at a Solution It seems...

Homework Statement Let ##G## be any group. Recall that the center of ##G##, or ##Z(G)## is ##\{ x \in G ~ | ~ xg = gx, ~ \forall g \in G\}##. Show that ##G / Z(G)## is isomorphic to ##Inn(G)##, the group of inner automorphisms of ##G## by ##g##. Homework EquationsThe Attempt at a Solution I am...

Homework Statement (part of a bigger question) For ##x,y \in \mathbb{R}^n##, write ##x \sim y \iff## there exists ##M \in GL(n,\mathbb{R})## such that ##x=My##. Show that the quotient space ##\mathbb{R}\small/ \sim## consists of two elements. Homework EquationsThe Attempt at a Solution Well...

$\tiny{242t.8.5.9}$ $\textsf{expand the quotient by}$ $\textbf{ partial fractions}$ \begin{align*}\displaystyle y&=\int\frac{dx}{9-25x^2} &\tiny{(1)}\\ \end{align*} $\textit{expand and multiply every term by $(3+5x)(3-5x)$}$ \begin{align*}\displaystyle...

Is there a general way to express, for instance, the coefficient of the order $x^j$ term in the expression $$\frac{\Sigma_{n}^{\infty}a_nx^n}{\Sigma_{m}^{\infty}b_mx^m}$$ ? Basically I am working with a quotient of two infinite power series and I want to know the term in this quotient that is...

Homework Statement Find the limit $$\lim_{x\to-\infty} \frac{\sqrt{9x^6 - x}}{x^3 + 9}$$ Homework Equations N/A The Attempt at a Solution To solve this, I start off by dividing everything by ##x^3##: Numerator becomes ##\frac{\sqrt{9x^6 - x}}{x^3} = \sqrt{\frac{9x^6 - x}{x^6}} = \sqrt{9 -...

Homework Statement I have been working with researching and writing a paper on the Freshman Dream Quotient Rule. This rule states , and I was wondering if anyone can come up with an example of 2 functions which make this work. Homework EquationsThe Attempt at a Solution I have looked at...