Product Definition and 1000 Threads

Hello all, I'm having trouble with proving that the derivative of f(x)*g(x) is f'(x)*g(x)+f(x)*g'(x). Now, I've already seen the actual proof, and I can understand its reasoning, but the first time I tried to prove without looking at the solution, this is what I wrote before I became rather...

In undergraduate abstract algebra we are not exposed to semi-direct products, so I was hoping someone could help me as I am doing some research in this area. I am familiar with the definitions of direct products and normal groups, and I know that a semidirect product is one where one of the...

In reading about the Tube Lemma, an example is given where the Tube Lemma fails to apply: namely, the euclidean plane constructed as R X R. The Tube Lemma does not apply here because R is not compact. The example given is as follows: Consider R × R in the product topology, that is the...

http://en.wikipedia.org/wiki/Torus ##S^1 \times S^1... \times S^1 \times S^1 ## is hypertorus. And what is ##S^2 \times S^2... \times S^2 \times S^2 ##?

Hello, why lim_{x\to (0)^{+}}e^{1/x}3x^2 = +\infty if lim_{x\to (0)^{+}}e^{1/x} = \infty and \lim_{x\to (0)^{+}}{3x^{2}} = 0 shouldn't it be +\infty * 0 ? I can't get it :( Thanks

Hello all, In a recent post, I discovered that when putting a product function in a fraction (using the \prod command), the indices are displayed to the right of the product function's symbol rather than below and above, which I find much more pleasing to the eye. I find that the same thing...

Hi, I have been learning about tensor products from Dummit and Foote's Abstract Algebra and I'm a little confused. I understand the construction of going to the larger free group and "modding out" by the relations that will eventually end up giving us module structure. But just in the...

Homework Statement The question I am trying to answer requires me to find the following: dN/dS ∝ S^−5/2/cosh(r/R) and I am giving the follwing equation in the question. A=4πR^2 sinh^2⁡〖(r/R)〗 The Attempt at a Solution Right I know how to get the S^-5/2 in the top half of the...

Hello MHB, I am just curious about how many in this forum got apple product :) I got one apple product which is Iphone 4s. Is there also anyone against apple?Regards, $$|\pi\rangle$$

Im having trouble understanding this property my book states that: a.(bxc) = b.(cxa) = c.(axb) it also states that a.(ax(anything)) = 0 I understand the second point and why that's true, what I don't understand is why a.(bxc) = b.(cxa) = c.(axb) is true If I name any 3 vectors a b...

Homework Statement Notice that ln [∏(k=1)^n a^k] = Ʃ_(k=1)^n * ln (a_k) I couldn't get the LaTeX right on this ^ But k=1 is below the product sign, and n is above. And (a^k) is the formula. From this, as well as some calculus, calculate that: lim as n->∞ ∏_(k=1)^n e^\frac{k^2}{n^3} For...

Homework Statement consider a vector z defined by the equation z=z1z2, where z1=a+jb, and z2=c+jd (j=complex: same as 'i'). (a) show that the length of z is the product of the lengths of z1 and z2. (b) show that the angle between z and the x-axis is the sum of the angles made by z1 and z2...

Hello, first post here. I am preparing for my Introductory Quantum Mechanics course, and in the exam questions, we are asked to use Ehrenfest's theorem to show that \frac{d}{dt}\langle \vec{r}\cdot \vec{p} \rangle = \langle 2T-\vec{r}\cdot \nabla V \rangle Now, from other results...

When one says that <\varphi|\psi> is the probability that \psi collapses to \varphi, does this "collapse" necessarily involve a measurement (so that one would have to find the implicit Hamiltonian)? Or does this just exist as part of the evolution of the wave function, perhaps the vacuum energy...

I have a spray sun protector, on it is a date that the product is good until 12 months after opening. However, there is no shelf date. Since it's a spray it has not been in contact with the outside environment, so why would it spoil? Should I continue using it? In other words, should I not...

How is computed the cross product of complex vectors? Let ##\mathbf{a}## and ##\mathbf{b}## be two vectors, each having complex components. $$\mathbf{a} = a_x \mathbf{\hat{x}} + a_y \mathbf{\hat{y}} + a_z \mathbf{\hat{z}}$$ $$\mathbf{b} = b_x \mathbf{\hat{x}} + b_y \mathbf{\hat{y}} + b_z...

Let n belongs to N, let p be a prime number and let $$Z/p^n Z$$denote the ring of integers modulo $$p^n$$ under addition and multiplication modulo $$p^n$$ .Consider two polynomials $$f(x) = a_0 + a_1 x + a_2 x^2 +...a_n x^n$$ and $$g(x)=b_0 + b_1 x + b_2 x^2 +...b_m x^m$$,given the coefficients...

Hi everyone, I'm trying to find a general rule that expresses the product of two rotation matrices as a new matrix. I'm adopting the topological model of the rotation group, so any rotation which is specified by an angle \phi and an axis \hat{n} is written R(\hat{n}\phi)= R(\vec{\phi})...

Problem: In Kleppner's book, Introduction to Mechanics, he states "By writing \vec{A} and \vec{B} as the sums of vectors along each of the coordinate axes, you can verify that \vec{A} \cdot \vec{B} = A_{x}B_{x} + A_{y}B_{y} + A_{z}B_{z}." He suggests summing vectors, but since the sum of two...

Hello folks! Just a concise introduction of myself before I get to the task at hand: I'm new to these forums although I have been surfing them frequently for the past 5 years! I am not a math major and quite frankly, my skills in the subject are limited. Be that as it may, my fascination for...

Consider ##\vec{a}=\begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix}## and ##\vec{b}=\begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix}##. Is any part of the following NOT true? $$\vec{a}\wedge\vec{b}=\frac{1}{2}(\vec{a}\otimes\vec{b}-\vec{b}\otimes\vec{a}) = \frac{1}{2}\begin{bmatrix} 0 &...

The cartesian product ∏X = Xi of a countable family {Xi} of regular spaces is zero-dimensional i f and only i f all spaces Xi , are zero-dimensional. I wonder if the countability assumption is just to ensure the regularity of the product space ,or it is crucial for the clopen basis. Thank's

Homework Statement A proof of equality between two traces of products of gamma matrices. Tr(\gamma^\mu (1_4-\gamma^5) A (1_4-\gamma^5) \gamma^\nu) = 2Tr(\gamma^\mu A (1_4-\gamma^5) \gamma^\nu) Where no special property of A is given, so we must assume it is just a random 4x4 matrix. 1_4...

Hello everyone, Say I have two irreducible representations \rho and \pi of a group G on vector spaces V and W. Then I construct a tensor product representation \rho \otimes \pi : G\to \mathrm{GL}\left(V_1 \otimes V_2\right) by \left[\rho \otimes \pi \right] (g) v\otimes w = \rho (g) v...

Hi everyone, I'm having some trouble with the concept of the direct sum and product of representations. Say I have two representations \rho_1 , \rho_2 of a group G on vector spaces V_1, V_2 respectively. Then I know their direct sum and their product are defined as \rho_1 \oplus \rho_2 : G...

Hello there, I posted the very same question before, nonetheless received no answers since -i presume- new posts arose and mine went on to the back of the data registry. I know many of you are math and physics experts, that's why I want you to please help me find out the integral of a product...

a + b + c = a * b * c where a, b, c are positive integers. I can think of only one solution to this. {1, 2, 3}. Is there any other solution to it? Can you prove or disprove?

for which value of a the expression write a*m*b+15*m+11*n+8 can be written into a product of something(with m)*something (with n) i can think how to find the product of this expression

Homework Statement Take P;Q and R three points of R 3 not on the same line. If a = OP , b = OQ and c = OR are the position vectors corresponding to the three points, show that a x b + b x c + c x a is perpendicular to the plane containing P;Q and R The Attempt at a Solution I don't...

Homework Statement Given the vectors: P = 8i +5j-Pzk m and Q = 3i -4j-2k m Determine the value of Pz so that the scalar product of the two vectors will be 60m2Homework Equations Sure seems like we will need to use the following equation: P * Q = |P| * |Q| * cos ∅ But I don't recall being able...

Hi everyone, I don't quite understand how tensor products of Hilbert spaces are formed. What I get so far is that from two Hilbert spaces \mathscr{H}_1 and \mathscr{H}_2 a tensor product H_1 \otimes H_2 is formed by considering the Hilbert spaces as just vector spaces H_1 and H_2...

Hello, some time ago I read that if we know the metric tensor g_{ij} associated with a change of coordinates \phi, it is possible to calculate the (Euclidean?) inner product in a way that is invariant to the parametrization. Essentially the inner product was defined in terms of the metric...

Homework Statement Calculate the uncertainty product ΔrΔp for the 1s electron of a hydrogen-like atom with atomic number Z. (Hint: Use <p> = 0 by symmetry and deduce <p^2> from the average kinetic energy) Homework Equations All I have is the wavefunction. For a 1s, it takes the form...

Hi everybody, I am just trying to find a decent identity that relates the sum $$\sum_{k=0}^{n}a_kb_k$$ to another sum such that ##a_k## and ##b_k## aren't together in the same one. If you don't know what I mean, feel free to ask. If you have an answer, please post it. Thanks in advance!

I am working on developing a new product but don't have an engineering/materials background so your input would be appreciated. The core of this product will be a wire/cable. It needs to have properties that make it very flexible and bendable and that can be somewhat weight bearing while...

Hello, let's assume we have an admissible change of coordinates \phi:U\rightarrow \mathbb{R}^n. I would like to know how the inner product on ℝn changes under this transformation. In other words, what is \left\langle \phi (u), \phi (v) \right\rangle for some u,v \in U ? I thought that...

Homework Statement Homework Equations r=\frac{S_{xy}}{S_{x} S_{y}} S_{x}=\sqrt{\Sigma(x-\bar{x})^2} The Attempt at a Solution I have calculated the value of Sx several times and I got 43.3 but the answer key said that Sx = 11.2 1. Is my formula correct? 2. If yes, is my...

Homework Statement Let A be the set that contains all rational numbers, but not zero. Let (a,b),(c,d) \in A×A. Let (a,b)\tilde{}(c,d) if and only if ad = bc. Prove that \tilde{} is an equivalence relation on A×A.Homework Equations The Attempt at a Solution The solution just needs to show...

Is the rule: P(AB I) = P(BA I) (which is used to derive Bayes rule) an axiom for probability? And if so, do you guys find it intuitive that it should hold. For instance consider a box with green and red beads. Do you think it is strictly obvious that the probability of getting red-green is...

Happy summer...ish! I am reading the lecture/presentation at: http://fias.uni-frankfurt.de/~brat/LecturesWS1011/Lecture5.pdf In slide two, they outline the Hartree product, but can anyone give a hint about how it's derived, or how to verify it? Put another way, can someone give me...

Homework Statement If x + v = u Prove x x v = u x v = x x u The Attempt at a Solution I don't even know where to start with this. I thought that magnitude of the resultant vector would have to be equal. So I started messing with each to see if I could find a pattern. x x v = | x||...

Homework Statement Hi Say I have an expression \arctan(a*b). Is there a product rule for \arctan that I can use to rewrite this? I tried the Wiki-page, but I couldn't find one there.

Hi everyone, I'm reading through tensor product spaces and one question really bogs me. Why is it that the total Hilbert space of a system composed of two independent subsystems is the tensor product of the Hilbert spaces of the subsystems? It is always posed, but I've never seen a proof...

This seems like a very basic question that I should know the answer to, but in my image processing class, my teacher explained that a basis set of images(matrices) are orthonormal. He said that the DOT product between two basis images (in this case two 2x2 matrices) is 0. so, for example...

Hi everyone, I have a project in which I have to introduce a new product into a companies manufacturing setup. The product has to include at least one of the following processes, metal forming, casting and at least one from the machining processes such as milling, drilling and turning. The...

EDIT: To moderators, I frequent this forum and the Calculus forum and I have accidently put this in Introductory Physics. Can it be moved? Sorry for inconveniences. Homework Statement 1)If ##G \cong H \times \mathbb{Z}_2, ## show that G contains an element a of order 2 with the property that...

"Sum to Product" Trigonometric identity does not work Hi, The identity [SIZE="3"]sin(u) + sin(v) = 2 * sin (\frac{u+v}{2}) * cos(\frac{u-v}{2}) http://en.wikipedia.org/wiki/List_of_trigonometric_identities#Product-to-sum_and_sum-to-product_identities Does not always work. I put the equation...

Hello. I'm just beginning my course in algebra. I've been reading Milne, Group Theory ( http://www.jmilne.org/math/CourseNotes/GT310.pdf page 29). I've found there a very nice proof of the fact that given two elements in a finite group, we cannot really say very much about their product's...

Homework Statement let \ell^{2} denote the space of sequences of real numbers \left\{a_{n}\right\}^{\infty}_{1} such that \sum_{1 \leq n < \infty } a_{n}^{2} < \infty a) Verify that \left\langle \left\{a_{n}\right\}^{\infty}_{1}, \left\{b_{n}\right\}^{\infty}_{1} \right\rangle =...

Hi all, I'm working on a math problem with a known answer - though I can't reproduce the maths. The problem is this: there is a random 3d vector of unit length with a uniform probability, \vec{v}, and a secondary unit vector \vec{u}. It is stated that: f = \int_{S^2}{| \vec{v} \cdot...