Product Definition and 1000 Threads

So I'm still playing with my waste oil heating system, and as I figured the oil spraying out of the burner doesn't burn 100% clean without a combustion chamber. I set it up outside with the burner pointing into a 12" section of 6" steel stove pipe with an elbow on the end, and it works perfect...

Hey! :o Let $A,B$ be sets, such that $A\times B=B\times A$. I want to show that one of the following statements hold: $A=B$ $\emptyset \in \{A,B\}$ I have done the following: Let $A$ and $B$ be non-empty set. Let $a\in A$. For each $x\in B$ we have that $(a,x)\in A\times B$. Since...

Hi, I'm trying to understand an outer product |1>_a<1| where |1>_a is the ket for one qubit (a) and <1| is the bra for another qubit. Does this make sense and is it possible to express it in terms of tensor products or pauli matrices?

I'm working out the quark loop diagram and I've drawn it as follows: where the greek letters are the Lorentz and Dirac indices for the gluon and quark respectively and the other letters are color indices. For this diagram I've written...

Summary: The Dot Product of a Vector 'a' and a Derivative 'b' is the same like the negative of the Derivative 'a' and the Vector 'b'. Hello, I have the following Problem. The Dot Product of a Vector 'a' and a Derivative 'b' is the same like the negative of the Derivative 'a' and the Vector...

Let c=ab. Let b be undefined (but finite). If a=0, is c undefined or does c=0?

How do I find the probabilty density function of a variable y being y=ab, knowing the probabilty density functions of both a and b? I know how to use the method to calculate it for a/b - which gives 1/pi*(a²/b²+1) - using variable substitution and the jacobian matrix and determinant, but which...

##p=\frac {RT} v;~p=p(T,v)~...1## ##v=\frac {RT} p;~v=v(T,p)~...2## ##T=\frac {pv} R;~T=T(p,v)~...3## ##Considering~eq.~1:## ##p=\frac {RT} v \Rightarrow (\frac {\partial p} {\partial v})_T=-\frac {RT} {v^2}## ##Considering~eq.~2:## ##v=\frac {RT} p \Rightarrow (\frac {\partial v}...

I see that when n is an even number, the product can be represented as ## \frac {2n} {(n+1)} ##. When n is an odd number, the denominator seems to be changing and I am not able to define an expression for it. How should I go about solving this? Thanks

There’s a old 2012 post on here “Why sine is used for cross product and cosine for dot product?” —there are a lot of great answers (which is how I came about this forum). After reading over the replies, it occurred to me: really it’s just because cosine is the “start” of a unit circle. Which...

Hey, sorry for the cluncky title. It was rathet difficult to summarise what I'm talking about here. I want to know if it's possible to define ##f(x)## and ##g(x)## in such a way that ##∫f(x)g'(x)dx## has no indefinite solution while ##∫f'(x)g(x)dx## does have an indefinite solution. Any help...

Starting with LHS: êi εijk Aj (∇xA)k êi εijk εlmk Aj (d/dxl) Am (δil δjm - δim δjl) Aj (d/dxl) Am êi δil δjm Aj (d/dxl) Am êi - δim δjl Aj (d/dxl) Am êi Aj (d/dxi) Aj êi - Aj (d/dxj) Ai êi At this point, the LHS should equal the RHS in the problem statement, but I have no clue where...

P&S had calculated this expression almost explicitly, except that I didn't find a way to exchange the $$\nu \lambda$$ indices, but I'm sure the below identity is used, $$ \begin{aligned}\left(\overline{u}_{1 L} \overline{\sigma}^{\mu} \sigma^{\nu} \overline{\sigma}^{\lambda} u_{2...

Dear all, I am trying to prove a simple thing, that if AxA = BxB then A=B. The intuition is clear to me. If a pair (x,y) belongs to AxA it means that x is in A and y is in A. If a pair (x,y) belongs to BxB it means that x is in B and y is in B. If the sets of all pairs are equal, it means...

Hello, I am calculating the krauss operators to find the new density matrix after the interaction between environment and the qubit. My question is: Is there an operational order between matrix multiplication and tensor product? Because apparently author is first applying I on |0> and X on |0>...

I am trying to make sense of the wikipedia article section regarding Cauchy product of several series. but am stuck right at the start because the notation used there is unfamiliar to me and not explained previously in the article. The commas in ##\Sigma a_1, k_1## etc. mean nothing to me. Am I...

Suppose I have a system of two (possibly interacting) spins of 1/2. Then the state of each separate spin can be written as a ##\mathbb{C}^2## vector, and the spin operators are made from Pauli matrices, for instance the matrices ##\sigma_z \otimes \hat{1}## and ##\hat{1} \otimes \sigma_z##...

From the properties of tensor product, ##H^{\otimes n} \cdot H^{\otimes n} =\left ( H_1 \cdot H_1 \right ) \otimes \left ( H_2 \cdot H_2 \right ) \otimes \cdots \otimes \left ( H_n \cdot H_n \right ) =I \otimes I \otimes \cdots \otimes I =I## where ##H_i## acts on the ##i^{th}## qubit. But I...

Let $$<x, x>=3x_{1}^2+2x_{2}^2+x_{3}^2-4x_{1}x_{2}-2x_{1}x_{3}+2x_{2}x_{3} $$ be a quadratic form in V=R, where $$x=x_{1}e_{1}+x_{2}e_{2}+x_{3}e_{3}$$ (in the base $${e_{1},e_{2},e_{3}}$$. Find the inner product corresponding to this quadratic form. Is this that easy that you have to change ''...

Hi! I'm given 2 points C(2;6) and D(0;10), a vector A with its components = (-3, 2). I'm asked to find the dot product between vector CD and an unknown vector K, knowing that K is perpendicular to A, same norm as A and with a negative x-component. I know that perpendicular means the dot...

I need help to know if I'm on the right track: Prove/Disprove the following: Let u ∈ V . If (u, v) = 0 for every v ∈ V such that v ≠ u, then u = 0. (V is a vector-space) I think I need to disprove by using v = 0, however I'm not sure.

Summary: I came across a question in my chemistry homework where i am supposed to write the balanced equation between ammonia and benzoic acid. I am not really good with chemistry but it's my last exam of chemistry ever in my high school experience, so i need to (and want to) get a good grade...

I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume II: Metric and Topological Spaces, Functions of a Vector Variable" ... ... I am focused on Chapter 11: Metric Spaces and Normed Spaces ... ... I need some help to confirm my thinking on Proposition...

https://www.physicsforums.com/attachments/8962 ok this is my overleaf homework page but did not do (c) and (d) this class is over but trying to do some stuff I missed. I am only auditing so I may sit in again next year...;) also if you see typos much grateful I don't see a lot of replies on...

I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume II: Metric and Topological Spaces, Functions of a Vector Variable" ... ... I am focused on Chapter 11: Metric Spaces and Normed Spaces ... ... I need some help to fully understand the proof of...

##U_1 \otimes U_2 = (1- i H_1 \ dt) \otimes (1- i H_2 \ dt)## We can write ## | \phi_i(t) > \ = U_i(t) | \phi_i(0)>## where i can be 1 or 2 depending on the subsystem. The ## U ##'s are unitary time evolution operators. Writing as tensor product we get ## |\phi_1 \phi_2> = (1- i H_1 \ dt) |...

Hi, I'm not a really mathematician...I've a doubt about the difference between a trivial example of fiber bundle and the cartesian product space. Consider the product space ## B \times F ## : from sources I read it is an example of trivial fiber bundle with ##B## as base space and ##F## the...

Problem: Prove that for any $x \in R^n$ and any $0<p<\infty$ $\int_{S^{n-1}} \rvert \xi \cdot x \rvert^p d\sigma(\xi) = \rvert x \rvert^p \int_{S^{n-1}} \rvert \xi_1 \rvert^p d\sigma(\xi)$, where $\xi \cdot x = \xi_1 x_1 + ... + \xi_n x_n$ is the inner product in $R^n$. Some thinking... I...

Hi, I'm currently working through a tensor product example for a two qubit system. For the expression: $$ \rho_A = \sum_{J=0}^{1}\langle J | \Psi \rangle \langle \Psi | J \rangle $$ Which has been defined as from going to a global state to a local state. Here $$ |\Psi \rangle = |\Psi^+...

Does anyone know where I can find the definition of ##\nabla \otimes \nabla f##? I tried googling this but nothing comes up. I know it will change depending on the coordinate system, so does anyone know the general definition OR a table for rectangular, spherical, cylindrical coordinates...

Hi I have used cross products thousands of time without really knowing what it actually does; I know how to compute it, but I don't feel like I understand it. Also, when it shows up in physics/kinematics contexts, it's only because the magnitudes of the vectors involved have to be multiplied...

Do we really need concept of cross product at all? I always believed cross product to be sort of simplification of exterior product concept tailored for the 3D case. However, recently I encountered the following sentence «...but, unlike the cross product, the exterior product is associative»...

Hello, I would like to write a product of two wave functions with a single ket. Although it looks simple, I do not remember seeing this in any textbook on quantum mechanics. Assume we have the following: ##\chi(x) = \psi(x)\phi(x) = \langle x | \psi \rangle \langle x | \phi \rangle## I would...

From the vector identity ##\nabla •fA=f(\nabla • A)+A•\nabla f## where f is a scalar and A is a vector. Now if f is an operator acting on A how does this formula change?? Like ##\nabla •[(v•\nabla)v]## where v is a vector

Evaluate: $$\prod_{n=1}^{\infty}\left(1+10^{-2^n}\right)$$

I am working through Tu's "An Introduction to Manifolds" and am trying to get an understanding of things with some simple examples. The definitions usually seem simple and understandable, but I want to make sure I can use them for an actual function. I've worked a few problems below that my...

In Loring W. Tu's book: "An Introduction to Manifolds" (Second Edition) ... Proposition 3.27 reads as follows: The above proposition gives the wedge product of k linear functions as a determinant ...Walschap in his book: "Multivariable Calculus and Differential Geometry" gives the definition of...

I know that a dot product of 2, 2 dimension vectors a, b = (ax * bx) + (ay * by) but it also is equal to a*bCos(θ) because of "projections". That we are multiplying a vector by the 'scalar' property of the other vector which confuses me because that projection is in the direction of the...

Hello everybody! I was studying the Glashow-Weinberg-Salam theory and I have found this relation: $$e^{\frac{i\beta}{2}}\,e^{\frac{i\alpha_3}{2} \begin{pmatrix} 1 & 0 \\ 0 & -1 \\ \end{pmatrix}}\, \frac{1}{\sqrt{2}}\begin{pmatrix} 0\\ v \\ \end{pmatrix} =...

I am reading Loring W.Tu's book: "An Introduction to Manifolds" (Second Edition) ... I need help in order to fully understand Tu's Proposition 3.21 ... ... Proposition 3.21 reads as follows: In the above proof by Tu we read the following: " ... ... ... ##= \sum_{ \sigma_{ k + l } } (...

I am reading Loring W.Tu's book: "An Introduction to Manifolds" (Second Edition) ... I need help in order to fully understand Tu's section on the wedge product (Section 3.7 ... ) ... ... The start of Section 3.7 reads as follows: In the above text from Tu we read the following: " ... ... for...

Homework Statement I have encountered this problem from the book The Physics of Waves and in the end of chapter six, it asks me to prove the following identity as part of the operation to prove that as the limit of ##W## tends to infinity, the series becomes an integral. The series involved is...

## \newcommand{\ihat}{\hat{\boldsymbol{\imath}}} \newcommand{\jhat}{\hat{\boldsymbol{\jmath}}} \newcommand{\khat}{\hat{\boldsymbol{k}}} ## Several times now I've seen the following technique for deriving the component form of the dot product. It always felt clean and simple until last night when...

Evaluate the following double sum of a product: $$\sum_{j=1}^{\infty}\sum_{n=1}^{\infty}\left(n\prod_{i=0}^{n}\frac{1}{j+i}\right)$$

Hi i have seen in abook the dot product defined as follows: Dot(A,B)=(1/4)[Norm(A+B)^2-Norm(A-B)^2] how this definition connect with the common one: Dot(A,B)=Sum(ai*bi) Thanks!

Homework Statement Prove that the product of any three consecutive integers is divisible by 6. Homework EquationsThe Attempt at a Solution This doesn't seem true to me for any 3 consecutive ints. For example, let a_0 = 0 a_1 = 1 a_2 = 2 3 is not divisible by six. Assuming they meant a_x...

Looking into the infinitesimal view of rotations from Lie, I noticed that the vector cross product can be written in terms of the generators of the rotation group SO(3). For example: $$\vec{\mathbf{A}} \times \vec{\mathbf{B}} = (A^T \cdot J_x \cdot B) \>\> \hat{i} + (A^T \cdot J_y \cdot B)...

Say that there is an object X = <ABC> = (A_1B_1C_1+A_2B_2C_2+...+A_NB_NC_N)/N Is there any way to say what X_A is? Or what exactly the A term in all of these terms contributed to X? Or is that info pretty much washed out in this type of ensemble average? Oh, and A, B and C are random...

Hello all. I'm using Griffiths' Introduction to Quantum Mechanics (3rd ed., 2018), and have come across what, on the face of it, seems a fairly straightforward principle, but which I cannot justify to myself. It is used, tacitly, in the first equation in the following worked example: The...

So say our inner product is defined as ##\int_a^b f^*(x)g(x) dx##, which is pretty standard. For some operator ##\hat A##, do we then have ## \langle \hat A ψ | \hat A ψ \rangle = \langle ψ | \hat A ^* \hat A | ψ \rangle = \int_a^b ψ^*(x) \hat A ^* \hat A ψ(x) dx##? This seems counter-intuitive...