quotient Definition and 356 Threads

Hi, Problem: Let X=\{x\times y|x^2+y^2\leq1\}, \mbox{ in } R^2. \mbox{ Let } X^{\star} \mbox{ be the partition of X consisting of all the one point sets } \{x\times y\}, x^2+y^2<1, \mbox{ along with the set } S^1=\{x\times y | x^2+y^2=1\}. \mbox{ Then it continues by saying...

Homework Statement Let G be the group { \begin{bmatrix}{a}&{b}\\{0}&{c}\end{bmatrix} | a, b, c are in Z_p with p a prime} Then let K = { \begin{bmatrix}{1}&{b}\\{0}&{1}\end{bmatrix} | b in Z_p} The map P: G --> Z*p x Z*p is defined by P(...

Homework Statement verify that R, the reals, quotiented by the equivalence relation x~x+1 is S^1 Homework Equations The Attempt at a Solution All i can think of is to draw a unit square and identify sides like the torus, but this would be using IxI, a subset of R^2, and gives a...

Can anyone explain, in detail, why/why not Z[X]/(2x) is isomorphic to Z/2Z? I know that every element in Z[x] can be written as a_0 + a_1 x + a_2 x^2 + ... with a_i in Z and only finitely many a_i's are nonzero. Now, does (2x) = (2, 2x, 2x^2,...)? Also, the quotient is "like" taking 2x=0, or...

Hi everyone, I have been trying to do this problem in both ways but I can't get the same answer the book says. This is the problem: x/ sqrt (x^2 +1) With quotient rule I got until the point I have [(x^2 +1)^1/2 - x^2/(x^2 +1)^1/2]/(x^2 +1) And with power rule I have [1/sqrt(x^2 +1)] -...

Homework Statement Let G be subgroup of Z^2=Z \times Z spanned by (4,2), (6,-12). Compute quotient Z^2/G. Homework Equations The Attempt at a Solution I was under the impression that I am supposed to write the vectors as columns in a matrix, and then compute the Smith-Normal...

Homework Statement Let R and S be rings. Show that \pi:RxS->R given by \pi(r,s)=r is a surjective homomorphism whose kernel is isomorphic to S. Homework Equations The Attempt at a Solution To show that \pi is a homomorphism map, I need to show that it's closed under addition and...

Homework Statement 3. (a) Let f(x) = ln(x^2-1), and [itex]g(x)=\frac{x}{\sqrt{2-x}}[/tex] (i) Find the natural domains of f, g, f + g, \frac{f}{g}, and \frac{g}{f} Homework Equations N/A The Attempt at a Solution I know that the natural domain of f(x) is x belongs to real...

I am still working on getting anything other than subscripts to post with my latex formatting, so for now I have posted a word document. Any help would be greatly appreciated, thanks. Joe

Homework Statement See attachment! 2. The attempt at a solution I multiplied by the LCD of (x+h-1)(x-1) and got 6h/h(x+h-1)(x-1) . I then got 6h/h^2+2hx-2h . However, my answer seems to be wrong. What is my error?

Homework Statement Find the derivative of \frac{sin x}{1 + cos x} Homework Equations Quotient rule \frac{gf' - fg'}{g^{2}} The Attempt at a Solution \frac{dy}{dx} = \frac{(1 + cos x)(\frac{d}{dx}(sin x)) - sin x(\frac{d}{dx}(1 + cos x)}{(1 + cos x)^{2}} simplify the...

Homework Statement Show that every element of the quotient group \mathbb{Q}/\mathbb{Z} has finite order but that only the identity element of \mathbb{R}/\mathbb{Q} has finite order. The Attempt at a Solution The first part of the question I solved. Since each element of...

"Group A is a quotient of Group B"? What does this phrase mean? I see it every now and again and can't figure it out. Are they say group B is the homomorphic image of group A? I'm familiar qith quotient groups, but with only groups A and B named, how would we know which quotient group of A...

I took a placement test and blew it away. (I tested into calculus, best possible placement for this test.) Everything was rather simple except for this problem which I cannot seem to get right. Can someone show me where I'm going wrong here?Homework Statement Use x2+2 in the Difference...

Homework Statement Given y = x^2 + 6 / x find x if dy/dx equals 0 zero Homework Equations dy/dx = v du/dx - u dv/dx / v^2 The Attempt at a Solution I have got as far as dy/dx = x^2 - 6 / x^2 However then zero must equal Sqrt 6, which 2.44... -...

Homework Statement if p(x) = f(x)/g(x) Prove that p'(x) = g(x) f '(x) - f(x) g '(x) / g(x)ˆ2 Homework Equations The Attempt at a Solution The proof goes like this in my book p(x + h) - p(x) / h = [ f(x+h)/ g(x+h) - f(x) / g(x) ] / h = f(x + h) g(x) - f(x) g(x +...

Homework Statement Hi all well basically i have finished off chain rule and right now i am going through product rule and quotient, as i was going through some questions , i understood the basic rule and so on, but why i don't get is, how do i figure which rule i need to apply given equation...

Can someone guide me on how to express (.a1a2a3a4a5) base n as a quotient of base n integers. There is a bar over a3a4a5.

Homework Statement Write the multiplication table of C_{6}/C_{3} and identify it as a familiar group. Homework Equations The Attempt at a Solution C_{6}={1,\omega,\omega^2,\omega^3,\omega^4,\omega^5} C3={1,\omega,\omega^2} The cosets are C3 and \omega^3C3 I just need help...

Homework Statement For a prime p and a polynomial g(x) that is irreducible in Z_{p}[X], prove that for any f(x) in Z_{p}[X] and integer k > 1, [f(x)]^{k} = [f(x)] in Z_{p}[X]/(g(x)). The Attempt at a Solution I realize this is an extension of Fermat's Little Theorem, however I cannot...

Hello everyone, first time poster here. I've been a lurker for about a week, but finally decided to join because I cannot for the life of me figure out this problem. Homework Statement Zach has trouble with the Quotient Rule; he thinks that d/dx (f(x)/g(x)) = f′(x)/g′(x). On his last...

I am trying to apply the Rayleigh quotient iteration to the matrix A= a 0 0 b where a is different from b (a and b are reals). Find the subsets S of R^2(Reals^2) having the property that the iteration applied to this matrix with initial guesses in S do not converge. Is S a set of...

The question is "without a calculator, long divide 425/836. Round to the nearest hundredth. Estimate your answer before starting." My attemp at a solution: Well, my estimate would be approximately one half, since 400 divided by 800 would be 0.5; however, I'm not sure how to long divide a...

Differentiate with respect to x; (using the quotient rule) 3/2x-1 (3 over 2x minus 1) dy/dx = (2x-1)(0) - (3)(2) / (2x-1)^2 dy/dx = -6/(2x-1)^2 but my book gives -2/(2x-1)^2 now, y = u/v and i take u = 3 and v = 2x-1. dy/dx = v(du/dx) -...

and simplify your awnser f(x)= (X+3) / (x+1) don't know how to do it, my attempt f(x+h)-f(x) /h ((x+3)/(x+1)+h)-(x+3/(x+1) /h bah how do i start and finish this?

im working on an equation using the quotient rule. this is the equation. 8(5+x)/16+X^2 and i have to find the f'= i know 16+x^2 = 2x im not sure about the 8(5+x) it can not be arctan(x) could you please help.

Hello everybody I would also like to solve the following problem using either the Chain,Product, or Quotient Rule but am unsure of the working stages to get to the given answers i) Find the equation of the tangent at the point with coordinates (1,1) to the curve with the equation...

Ok so you can't apply the quotient criteria to the harmonic series because: lim_{k\to \infty}|\dfrac{a_{k+1}}{a_k}| applied to the harmonic series: lim_{k\to \infty}|\dfrac{1/(k+1)}{1/k}| = lim_{k\to \infty}|\dfrac{k}{k+1}| < 1 which does fullfill the quotient criteria, yet the...

Notations: V denotes a vector space S denotes a subspace of V V/S denotes a quotient space V\S denotes the complement of S in V Question: If {s1, ... , sk} is a basis for S, how to find a basis for V/S? I realize that the basis of V\S may determine the basis of V/S, but I don't know...

Homework Statement Let f(x) = x2 + 1 in Z3[x]. Find the order of the quotient ring Z3[x]/<f>. Homework Equations The Attempt at a Solution Note Z3 is a field. Then Z3[x] is euclidean domain. Then for any polynomial g(x) can be written as g(x) = p(x).(x2+1) + r(x) where...

Prove that b_{ijkl}=\int_{r<a} dV x_i x_j \frac{\partial^2}{\partial_k \partial_l} (\frac{1}{r}) where r=|x| is a 4th rank tensor. i've had a couple of bashes and got nowhere other than to establish that its quotient theorem. can i just pick a tensor of rank 3 to multiply it with or...

Find, up to isomorphism, all possible quotient groups of D6 and D9, the dihedral group of 12 and 18 elements. First of all, I don't understand the question by what they mean about "up to isomorphism." Does this mean by using the First Isomorphism Theorem? Also does this question imply that...

Homework Statement Given a four degree of freedom system that consists of four carts. The four carts each have mass m=1, and they are connected by three springs of constant k=4, 1, 1 respectively. Let x, y, z, and w be the displacement from equilibrium of the four carts, relative to the...

I don't take a Calculus class(I'm learning on my own), but I'm just curious as to what are the steps to solving the following equations. Homework Statement These are the following problems that I'm having trouble solving. y(x) = cos(x) y(x) = \sqrt{x} y(x) = sin(x) y(x) =...

Homework Statement In Z4 x Z4, find two subgroups H and K of order 4 such that H is not isomorphic to K, but (Z4 x Z4)/H isomorphic (Z4 x Z4)/K Homework Equations The Attempt at a Solution I know (Z4 x Z4) has twelve elements (0,0), (1,0), (2,0), (3,0), etc. I can generate subgroups of...

Homework Statement I would like to know...when trying to take the derivative of a function with a fraction in it ... should I always turn it into a product and use the product rule, thereby dropping the quotient rule most of the time? Or is the quotient rule needed more so in some cases...

Simplify \frac{sinx + tanx}{cscx + cotx} I start with \frac{sinx + tanx}{cscx + cotx} = \frac{sinx + \frac{sinx}{cosx}}{\frac{1}{sinx} + \frac{cosx}{sinx}} At this point I am stuck, I cannot see how we then get from \frac{sinx + \frac{sinx}{cosx}}{\frac{1}{cosx} + \frac{cosx}{sinx}} =...

Homework Statement Let A be a symmetric n x n - matrix with eigenvalues and orthonormal eigenvectors (\lambda_k, \xi_k) assume ordening: \lambda_1 \leq...\leq \lambda_n We define the rayleigh coefficient as: R(x) = \frac{(Ax)^T x}{x^T x} Show that the following constrained problem...

Homework Statement Prove that is m, n, and d are integers and d divides (m-n) then m mod d = n mod d. Homework Equations Quotient Remainder Theorem: Given any integer n and positive integer d, there exists unique integers q and r such that n=dq + r and 0\leqr<d and n mod d = r. The...

Homework Statement I'm having a hard time understanding quotient rings. I think an example would help me best understand them. For example, how does the ring structure of \mathbb{F}_{2}/(x^4 + x^2 + 1) differ from that of \mathbb{F}_{2}/(x^4 + x + 1)? Homework Equations The...

Domain Functions: f(x) = 4sq.root(1-x2), the 4 is on the outside f(s) = sq.root(s -1)/s-4 f(x) = x-4/sq.root x I have absolutely no clue how to go about doing any of these, I take notes but I cannot piece it together; I have never felt so helpless at anything. This isn't homework I...

what is quotient rule for higher order derivatives ? i mean the one analogous to http://en.wikipedia.org/wiki/Leibniz_rule_%28generalized_product_rule%29" .

Homework Statement Find: Lim | x2+x-12 |-8 / (x-4) x --> 4 Homework Equations The Attempt at a Solution My answer is 9. It it right ? or there is not a limit for F(x) when x --> 4

Is a projection a quotient map? I think a quotient map is an onto map p:X-->Y (where X and Y are topological spaces) such that U is open/closed in Y iff (p)-1(U) is open/closed in X. And a projection is a map f:X-->X/~ defined by f(x)=[x] where [x] is the equivalent class (for a...

Hello everybody, I'm a bit stuck here. I have a problem tha goes like this: Let R be a principal ideal domain (PID). Let D a subset of R which is multiplicatively closed. Show that the ring of quotients D^(-1)R is a PID too. I've tried several different ways but I couldn't get to the...

1. Show that every element of the quotient group G = Q/Z has finite order. Does G have finite order? he problem statement, all variables and given/known data [b]2. This is the proof The cosets that make up Q/Z have the form Z + q, where q belongs to Q. For example, there is a...

Can anyone post me a clear example of how to compute the quotient space U/V from a vector space U and subspace V? I've seen many formal definitions but I'm a little stuck on practical use. I'm particularly interested in an example that shows how U and V being over the reals (for example) can...

Find the gradient of F(s,t) = f(x(s,t), y(s,t)) where f(x,y) = y/x x = s^2 + t^2 y = s^2 - t^2. I'm not sure how to even start the problem. Could someone point me in the right direction?

i am confused about how to find the subgroup of a quotient group given a generator. for example, a lot of problems give as the group Z/nZ with n very large. how do you find the subgroup given a generator? thanks!

Homework Statement Using difference quotient I am trying to find f '(0) for 2^x. Basically my question is a questions of algebra but I will show you what I have done thus far. the limit is as x -> 0 \frac{2^x - 2^0}{x - 0} \frac{2^x - 1}{x} So my question is what can I do to get x...