Product Definition and 1000 Threads

Hello MHB. I need help with the follwoing: Evaluate $\displaystyle P=\prod_{j=1}^{1004}\cos\left(\frac{2\pi j}{2009}\right)$. I think it would be easier to first evaluate $\displaystyle Q=\prod_{j=1}^{2008}\cos\left(\frac{2\pi j}{2009}\right)$. Knowing the value of $Q$ we can easily find the...

momentum, position vector dot (scalar) product "action" Hello, I was playing with single mass point classical mechanics, when I realized that the dot product of the position vector and momentum vector, p.r , has action dimension. Furthermore, its time derivative, d/dt(p.r) = F.r + p.v, has...

Integrating a product of two functions - one "lags" the other I am wondering if there is a way to integrate the following function without first expanding the brackets: \int\limits_{x1}^{x2} x^2\left(x-a\right)^2\,dx The idea behind the question is a bit more complex than I am letting on...

Why does the cross product produce a vector and why is that vector perpendicular to the other vectors? I understand how to calculate a cross product, but why for instance is the cross products of two vectors another vector that is perpendicular to it. Can you prove or explain this to me in...

Homework Statement A=(x,3,1) ,B=(x,-x,2) Determine the value of x if the vector perpendicular to A and B is given by C=(10,-4,-4) Homework Equations The Attempt at a Solution Find A cross B , let A cross B be D . Then D cross C = zero (since they are perpendicular to both A and...

Hello, It may be trivial to many of you, but I am struggling with the following integral involving two spheres i and j separated by a distance mod |rij| ∫ ui (ρ).[Tj (ρ+rij) . nj] d2ρ The integration is over sphere j. ui is a vector (actually velocity of the fluid around i th...

Basically I don't know anyone in real life that can help me with this, so I need help checking to see if my answers are correct :) PART A 11) Find the vertex of f(x) = -2x^2 - 8x + 3 algebraically. My Answer: (-2,0) 12) Multiply and simplify: (6 - 5i) (4 + 3i) My Answer: 39 - 2i

Homework Statement Does the Cauchy Schwarz inequality hold if we define the dot product of two vectors A,B \in V_n by \sum_{k=1}^n |a_ib_i| ? If so, prove it. Homework Equations The Cauchy-Schwarz inequality: (A\cdot B)^2 \leq (A\cdot A)(B\cdot B) . Equality holds iff one of the vectors...

Is this correct in infinite dimensional Hilbert spaces? ## (\hat{A}_1 \otimes \hat{A}_2)^{-1}=\hat{A}^{-1}_1 \otimes \hat{A}^{-1}_2 ## ## (\hat{A}_1 \otimes \hat{A}_2)^{\dagger}=\hat{A}^{\dagger}_1 \otimes \hat{A}^{\dagger}_2 ## ## (\hat{A}_1 +\hat{A}_2) \otimes \hat{A}_3=(\hat{A}_1 \otimes...

a and b are two vectors and x is the angle between them. ||axb|| = ||a||||b||sinx ------(1) ||axb|| = ||a||||b||||sinx|| ------(2) which one is correct? why?

Homework Statement There are two locations for the lens along the optical bench that will focus an image on the screen. Find one of these locations. Once you have the image in sharp focus, take measurements for: the object distance, do, the image distance, di, and the height of the image, hi...

1) Show that for $n >1$, $\displaystyle \prod_{k=1}^{\infty} \left( 1- \frac{z^{n}}{k^{n}} \right) = \prod_{k=0}^{n-1} \frac{1}{\Gamma\left[ 1-\exp (2 \pi i k/n) z\right]}$.2) Use the above formula to show that $ \displaystyle \prod_{k=1}^{\infty} \left(1- \frac{z^{2}}{k^{2}} \right) =...

Homework Statement Let V be the real vector space of all real symmetric n × n matrices and define the scalar product of two matrices A, B by (Tr (A) denotes the trace of A) Show that this indeed fulfils the requirements on a scalar product. Homework Equations Conditions for a scalar...

Homework Statement Consider the vector space of continuous, complex-valued functions on the interval [−∏, ∏]. Show that defines a scalar product on this space. Are the following functions orthogonal with respect to this scalar product? Homework Equations The Attempt at a...

I have always wondered: Is the product rule and addition rule for that matter axioms of the probability theory or can they actually be proven from more general principles? The reason I ask is, and it might be a bit silly, that I have always thought I missed out on something in probability...

Consider 2 atomic orbitals with wave function a: σ(r), b: μ(r) in a diatomic molecules. σ(r) (or μ(r)) is localized around an atom a (or b) and is relevant for the discussion of the molecular orbital. These orbitals are orthogonal and normalised. The creation operators are x, y and vacuum, |0>...

a =(x,y), b =(h,k) a cross b =? I have idea what to type on google. Is that doing like matrices , a cross b = xk-hy? thanks.

Let R be a UFD and K be its field of fractions. Let f(x) be in R[x] and f(x) = a(x)b(x) where a(x), b(x) are in K[x]. Show that there exists a c in K such that c*a(x) and c-1*b(x) are both in R[x] and such that f(x) = (c*a(x))(c-1*b(x)) in R[x]. I have been stuck on this for a while now as I...

Homework Statement Write each polynomial as the product of it's greatest common monomial factor and a polynomial Homework Equations 8x^2+12x 6a^4-3a^3+9a^2 The Attempt at a Solution 4x(2x+3) a^2(6a^2-3a+9) I really don't understand what it's asking me for, how can I factor this...

Homework Statement L(θ) = ∏(θ/(2√xi)*e^(-θ√xi)),i=1, n Homework Equations The Attempt at a Solution -> θ2∏(1/(2√xi)*e^(-θ√xi)) taking natural log of both sides lnL(θ) = nlnθ + ln∏(1/(2√xi)*e^(-θ√xi)) = nlnθ + Ʃln(1/(2√xi)*e^(-θ√xi)) Ok so from what I understand the...

Homework Statement Let A be a vector perpendicular to every vector X. Show that A = O Edit: it is O not 0. (OH not zero) haHomework Equations So, we know if A and X are perpendicular then A(dot)X = 0 I see no reason why A would have to be equal to 0. Could X (not equal) 0? Could it be...

Amazon.com (USA) is quietly deleting all customer contributed product photos from its website. The deletion is scheduled to be complete by Aug. 31. If you browse the site you'll notice the option to upload customer images is gone from most of the pages. (Meanwhile, Amazon has announced that...

Hi, I was reading vector mechanics by beer and jhonston. I came across the equation wherein the cross prodcut of two vectors P and Q is given. It says P x Q = P x Q` . I am not bale to understand how dis is possible. Because as the vector Q changes even the angle teetha will change then how...

Understanding the use of Pythagorean theorem for the length and square of a vector and how this is the dot product of a vector with itself is no problem. I'm trying to look inside the meaning of the dot product of two different vectors and understand it. I can also accept (just following the...

I would like to take divergence of the following expression ∇.((xi × xj × xk)/r^3), which is a triadic. here × denote a dyadic product and r=mod(r vector) and xi, xj and xk are the components of r vector. So, the above eq. can also be written as ∇.((xixjxk ei×ej×ek)/r^3), where ei...

so, the magnitude of the cross product represents the area or volume enclosed by any 2 or 3 vectors respectively, but what does the direction represent? i get that the general direction is one that is perpendicular to all the vectors, but what does the actual direction represent? ie, why the...

I would like to know if the Jordan Product carries along with it the same properties as that of the cross product (i.e. associativity, commutable, left/right distributive)? If you've taken LA, I'm sure you know that professors require us to complete a project and I've chosen the Jordan Product...

Is there atopological space X such that XxX (the product space) is homeomorphic to the unit circle in the plane

I have two real symmetric matrices A and B with the following additional properties. I would like to know how the eigenvalues of the product AB, is related to those of A and B? In particular what is \mathrm{trace}(AB)? A contains only 0s on its diagonal. Off diagonal terms are either 0 or...

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 Zdenote 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 are in Z/p^nZ...

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 &...