Product Definition and 1000 Threads

Hello Physics Peeps, It just came up in the notes for my electrodynamics class that an electrons charge squared can be expressed as the radius times the mass times the speed of light squared. e^2 = m_er_ec^2 I don't understand the motivation for doing this. I've tried to search for other...

Let \mathbb{F} be an arbitrary field, and let a,b\in\mathbb{F}^3 be vectors of the three dimensional vector space. How do you prove that if a\times b=0, then a and b are linearly dependent? Consider the following attempt at a counter example: In \mathbb{R}^3 \left(\begin{array}{c} 1 \\ 4 \\ 2...

What has done here in the second line of the proof for product rule?, from Mathematical methods for physicists from Riley, Hobson they defined f(x)=u(x)v(x) and these steps are given, I have no idea how to proceed further please help me.

Homework Statement The product formed in the reaction of SOCl2 with white phosphorus is 1. PCl3 2. SO2Cl2 3. SCl2 4. POCl3 Homework Equations NA The Attempt at a Solution I can google that but I want to know that how can we know it intuitively or by ourselves? It was asked in a test and in...

I was wondering what the physical insight is of integrating a product of two functions. When we do that for a Fourier transform, we decompose a function into its constituent frequencies, and that's because the exponential with an imaginary x in the transform can be seen as a weighting function...

Long story short, I currently work in digital product management, I am successful and it is lucrative. However, I never finished my college degree (originally business focused), and at 34 years old I would be starting over at this point. I was widowed a few years ago and am a single mom to a...

Homework Statement Prove that for any three vectors ##\hat a, \hat b ## and ## \hat c##, ##\hat a \cdot (\hat b \times \hat c)## = ##(\hat a \times \hat b) \cdot \hat c ## Homework Equations [/B] ## \hat i \cdot \hat i = \hat j \cdot \hat j = \hat k \cdot \hat k = (1)(1)\cos(0) = 1 ## ##...

Homework Statement Which of the following expressions represents the solubility product for Cu(OH)2? (A) Ksp=[Cu2+][OH-]2 (B) Ksp=[Cu2+]2[OH-] (C) Ksp=[Cu2+]2[OH-]2 (D) Ksp=[Cu2+][OH-] Homework Equations Ksp= [A][ B] The Attempt at a Solution Okay, so I understand equilibrium expressions and in...

What do you call the final product or products that exit from a reactor

Okay I'm having a little trouble understanding a section of this proof about the product of the gradients of perpendicular lines given in my textbook. I'm going to type the proof out but there will be a link at the bottom to an online version of the textbook so you can see the accompanying...

If ##\sigma## is an affine paramter, then the only freedom of choice we have to specify another affine parameter is ##a\sigma+b##, a,b constants. [1] For the tangent vector, ##\xi^{a}=dx^{a}/du##, along some curve parameterized by ##u## My book says that ' if ##\xi^{a}\xi_{a}\neq 0##, then by...

Can anyone tell me why?

Homework Statement [/B] hi could some body please help me factorise this please ? any chance of a few stages would be much appreciated Homework EquationsThe Attempt at a Solution my attempt , but my solutions say otherwise ? [/B]

Homework Statement Claim: If ##A \in \mathcal{M}_n (\mathbb{C})## is arbitrary, and ##D## is a matrix with ##\beta## in its ##(i-j)##-th entry, and ##\overline{\beta}## in its ##(j-i)##-th, where ##i \ne j##, and with zeros elsewhere, then ##Tr(AD) = a_{ij} \beta + a_{ji} \overline{\beta}##...

(I hope this post goes in this part of the forum) Hi, I was wondering if someone could help me with the following: I have a (1+1) scalar field decomposed into two different sets of modes. One set corresponds to a Minkowski frame in (t,x) coordinates, the other to a Rinder frame in conformal...

Homework Statement Given a×b=-i-j+3k and c×a=2i-3j+k, find a×(a-2b+c) Homework Equations Cross product (DONE WITHOUT MATRICES). The Attempt at a Solution a[/B]×b=c=-(b×a)is all I'm getting to at this point

Homework Statement [/B] Sakurai problem 1.20: find the linear combination of spin-up and spin-down S_z eigenkets that maximizes the uncertainty product \langle(\Delta S_x)^2\rangle\langle(\Delta S_y)^2\rangle. Homework Equations [/B] In general, we can write a normalized spin-space ket as...

Hi, I want to show that the Trace of the Product of a symetric Matrix (say A) and an antisymetric (B) Matrix is zero. $So\: (AB)_{ij}=\sum_{k}^{}{a}_{ik}{b}_{kj} $ $and\: Tr(AB)=\sum_{i=j}^{}(AB)_{ij}=\sum_{i}^{}\sum_{k}^{}{a}_{ik}{b}_{ki} $ $because\:A\:is\:symetric, \: {a}_{ik}=...

Assume that G is some group with two normal subgroups H_1 and H_2. Assuming that the group is additive, we also assume that H_1\cap H_2=\{0\}, H_1=G/H_2 and H_2=G/H_1 hold. The question is that is G=H_1\times H_2 the only possibility (up to an isomorphism) now?

Homework Statement By considering A x (B x A) resolve vector B into a component parallel to a given vector A and a component perpendicular to a given vector A. Homework Equations a x (b x c) = b (a ⋅ c) - c (a ⋅ b) The Attempt at a Solution I've applied the triple product expansion and...

Hi, I am studying Sean Carroll's "Lecture notes on General Relativity". In the second chapter he identifies the volume element d^nx on an n-dimensional manifold with dx^0\wedge\ldots\wedge dx^{n-1}. He then claims that this wedge product should be interpreted as a coordinate dependent object...

Homework Statement Let f_1,f_2\colon\mathbb{R}^m\to\mathbb{R} and a cluster point P_0\in D\subset\mathbb{R}^m (domain) Prove that \lim_{P\to P_0} f_1(P)\cdot f_2(P) = \lim_{P\to P_0} f_1(P)\cdot\lim_{P\to P_0} f_2(P) Homework EquationsThe Attempt at a Solution Let \begin{cases} \lim_{P\to...

Homework Statement Need to prove that: (v⋅∇)v=(1/2)∇(v⋅v)+(∇×v)×v Homework Equations Vector triple product (a×b)×c=-(c⋅b)a+(c⋅a)b The Attempt at a Solution I know I could prove that simply by applying definitions directly to both sides. I haven't done that because that is tedious, and I...

I want to find the solution of vector X. I am using text from Alan F. Beardon Algebra and Geometry as attached. I don't know how the solution is derived for the following equation. ## x + (x × a) = b ## The second solution when ## a \times b \neq 0 ## then X cannot be b. Is it possible to...

How do you show that $$\frac{1}{z}\prod_{n=1}^\infty \frac{n}{z+n}(\frac{n+1}{n})^z$$ is meromorphic? Any hints would be helpful, I'm having trouble bounding the functions and their logarithms. This is exercise XIII.3 problem 15 in Gamelin's Complex Analysis.

Please see the attached picture. I'm not sure which hydrogen would be anti-periplanar. Any help would be appreciated. Thanks!

Hello everyone, I have thi doubt: If I have a state, say psi1, associated with the energy eigenvalue E1, the integral over a certain region gives me the probability of finding the particle in that region with the specified energy E1. Now if I put an operator between the states I obtain its mean...

In biology i have studied that in plants some secondary product examples are cuticle , lignin etc ... can you tell me why it is called secondary products ... thank you.

In a book I was reading, it says F=mv'=P' so dot producting on both sides with v F ⋅ v = mv ⋅ dv/dt = 1/2 m d(v2)/dt = d(1/2 m v^2)/dtI really don't get how v ⋅ dv/dt = 1/2 d(v2)/dt. I have seen few threads and they say it's about product rule, but they don't really explain in detail. Could...

Hi all, I am very confused on how to define the vector product or cross product in a physical sense. I know the vector product is a psuedovector, and that it is the area of a parallelogram geometrically. However, I know it used used to describe rotation in physics. As with torque, magnetism and...

Homework Statement Find the set of points of M such that: AM x BC=AM x AC (Vectors) The Attempt at a Solution [/b] AM x (BM+MC) =AMx(AM+MC) AMxBM+AMxMC=AMxAM +AM x MC Then AMxBM=0 MA X MB=0 I am new to this lesson and this is my first time i solve such a question and i had no idea...

Hi all, I am a final year maths student and am doing my dissertation in the finite element method. I have gotten a little stuck with some parts though. I have the weak form as a(u,v)=l(v) where: $$a(u,v)=\int_{\Omega}(\bigtriangledown u \cdot\bigtriangledown v)$$ and $$l(v)=\int_\Omega...

$\d{}{x}{3}^{x}\ln\left({3}\right)=$ I tried the product rule but didn't get the answer😖

Hi, I have following problem of double dot product (\vec a \cdot \vec b)(\vec a^* \cdot \vec c), and I have expected rusult |a|^2(\vec b \cdot \vec c), but I don't know if it is the exactly result (I am unable to find any appropriate identity or proove it), or it is just an approximation...

Mod note: Member warned about posting with no effort. 1. Homework Statement Expand to the general case to explore how the cross product behaves under scalar multiplication k (a x b) = (ka) x b = a x (kb). The Attempt at a Solution would this be the right general case to portray the situation?

I am trying to prove the following. Let $V_1, \ldots, V_k$ be finite dimensional vector spaces over a field $F$. There is a natural isomorphism between $V_1^*\otimes\cdots\otimes V_k^*$ and $\mathcal L^k(V_1, \ldots, V_k;\ F)$. Define a map $A:V_1^*\times\cdots\times V_k^*\to \mathcal L^k(V_1...

Hey guys, So consider the following product of matrices: (p_{1}^{\mu}\cdot p_{1}^{\prime\nu} -(p_{1}\cdot p_{1}')\eta^{\mu\nu}+p_{1}^{\nu}p_{1}^{\prime\mu})(p_{2\mu}p_{2\nu}'-(p_{2}\cdot p_{2}')\eta_{\mu\nu}+p_{2\nu}p_{2\mu}') where eta is the Minkowski metric. I keep getting 2(p_{1}\cdot...

(All vector spaces are over a fixed field $F$). Universal Property of Tensor Product. Given two finite dimensional vector spaces $V$ and $W$, the tensor product of $V$ and $W$ is a vector space $V\otimes W$, along with a multilinear map $\pi:V\times W\to V\otimes W$ such that whenever there is...

Hi, I that <I|M|J>=M_{I}^{J} is just a way to define the elements of a matrix. But what is |I>M_{I}^{J}<J|=M ? I don't know how to calculate that because the normal multiplication for matrices don't seem to work. I'm reading a book where I think this is used to get a coordinate representation of...

Hi guys, So I've got a real scalar field which is the sum of the positive frequency part and negative frequency part: \phi(x)=\phi^{(+)}(x)+\phi^{(-)}(y) and I'm looking at the time-ordered product: T(\phi(x)\phi(y))=\theta(x^{0}-y^{0})\phi(x)\phi(y)+\theta(y^{0}-x^{0})\phi(y)\phi(x) for...

Homework Statement [/B] Consider a real free scalar field Φ with mass m. Evaluate the following time-ordered product of field operators using Wick's theorem: ∫d^4x <0| T(Φ(x1)Φ(x2)Φ(x3)Φ(x4)(Φ(x))^4) |0> (T denotes time ordering) Homework Equations Wick's theorem: T((Φ(x1)...Φ(xn)) = ...

Homework Statement A and B are matrices and x is a position vector. Show that $$\sum_{v=1}^n A_{\mu v}(\sum_{\alpha = 1}^n B_{v\alpha}x_{\alpha})=\sum_{v=1}^n \sum_{\alpha = 1}^n (A_{\mu v} B_{v\alpha}x_{\alpha})$$ $$= \sum_{\alpha = 1}^n \sum_{v=1}^n(A_{\mu v} B_{v\alpha}x_{\alpha})$$ $$=...

I want to learn clifford and grassmannian algebras. I need to be taken from mostly a beginners point, and from a place of matrices only in general terms, and years since use. ANybody up for it? I am a software developer, so not at the bottom of any learning curve.

I've attached an image of part a of the question to this thread. My question is this (the solution to these former homework problems are posted to help us study for exam, which is why know this already): The angle between the two velocity vectors is determined to be pi/2. How? I know that dot...

Hello, I have this exercise that I can't get the right answer. I have to find derivative of g(x)= (4${x}^{2}$-2x+1)${e}^{x}$ So, what is did is g$^{\prime}$=(8x-2)${e}^{x}$+(4${x}^{2}$-2x+1)${e}^{x}$ My Prof said it is wrong... I am not sure if I have to multiply the brackets or what I did...

1]express j! in ∏ notation Are they just wanting something like j! + (j-1)! + (j-2)! +(j-3)!...?

I'm trying to re-derive a result in a paper that I'm struggling with. Here is the problem: I wish to calculate (\nabla \otimes \nabla) h where \nabla is defined as \nabla = \frac{\partial}{\partial r} \hat{\mathbf{r}}+ \frac{1}{r} \frac{\partial}{\partial \psi} \hat{\boldsymbol{\psi}} and...

I know for two linear operators $$H_1, H_2$$ between finite dimensional spaces (matrices) we have the relations (assuming their adjoints/inverses exist): $$(H_1 H_2)^* = H_2^* H_1^*$$ and $$(H_1 H_2)^{-1} = H_2^{-1} H_1^{-1}$$ but does this extend to operators in infinite dimensions? Thanks.

I am trying to work out with Young graphs the tensor product of: \bar{3} \otimes \bar{3} The problem is that I end up with: \bar{3} \otimes \bar{3} = 15 \oplus 6 \oplus 3 \oplus 3 Is that correct? It doesn't seem correct at all (dimensionally speaking I should have taken something like...

I am trying to solve for the energy of 2 non-interacting identical particles in a 1D infinite potential well. I want to do it as much "from scratch" as possible, making sure I fully understand every step. H = -ħ2/2m * (∂2/∂x12 + ∂2/∂x22) Hψ=Eψ ∂2ψ/∂x12 + ∂2ψ/∂x22 = kψ, where k=-2mE/ħ2 I got...