Product Definition and 1000 Threads

It seems there should be a list of tensor identities on the internet that answers the following, but I can't find one. For tensors in ##R^4##, ##S = S_\mu{}^\nu = S_{(\mu}{}^{\nu)}## is a symmetric tensor. ##A = A_{\nu\rho\sigma}= A_{[\nu\rho\sigma]}## is an antisymmetric tensor in all...

Let's say we have: \vec{E}=E_x\vec{i}_x+E_y\vec{i}_y+E_z\vec{i}_z and \vec{B}=B_x\vec{i}_x+B_y\vec{i}_y+B_z\vec{i}_z and the Lorentz Force 0=q(\vec{E}+\vec{v}X\vec{B}) which due to \vec{E}X\vec{B}=\vec{B}X(\vec{v}X\vec{B})=vB^2-B(\vec{v}\cdot \vec{B}) and transverse components only...

From a textbook. proof that the scalar product ##A\centerdot B## is a scalar: Vectors A' and B' are formed by rotating vectors A and B: $$A'_i=\sum_j \lambda_{ij} A_j,\; B'_i=\sum_j \lambda_{ij} B_j$$ $$A' \centerdot B'=\sum_i A'_i B'_i =\sum_i \left( \sum_j \lambda_{ij} A_j \right)\left( \sum_k...

Homework Statement https://www.dropbox.com/s/8l90hahznjlv9d0/vector%20problem.png?dl=0 Homework Equations Dot and Cross product The Attempt at a Solution although I know the dot and cross product, I'm not sure what I'm being asked or how to proceed? any help?[/B]

Hi all. I'm struggling with taking dot products between vectors in spherical coordinates. I just cannot figure out how to take the dot product between two arbitrary spherical-coordinate vectors ##\bf{v_1}## centered in ##(r_1,\theta_1,\phi_1)## and ##\bf{v_2}## centered in...

Suppose we have the sets $A=\left\{2,3\right\}$ and $B=\left\{5\right\}$, then $A$ X $B$ is defined as $\left\{(x,y)|x \in A, y\in B\right\}=\left\{(2,5), (3,5)\right\}$. But what happens when $A$ contains elements that are not in $\Bbb{R}$? Example: $A=\left\{(2,3),(3,4)\right\}\subset...

Homework Statement Prove that $$log_{b}(xy)=log_{b}x+log_{b}y.$$ Homework Equations Let $$b^{u}=x,b^{v}=y.$$ Then $$log_{b}x=u,log_{b}y=v.$$ The Attempt at a Solution I'm afraid I've been using circular reasoning to prove this. I can get this to a point where I have...

Two lines A and B. The angle between them is θ, their direction cosines are (α,β,γ) and (α',β',γ'). Prove, ON GEOMETRIC CONSIDERATIONS: ##\cos\theta=\cos\alpha\cos\alpha'+\cos\beta\cos\beta'+\cos\gamma\cos\gamma'## I posted this question long ago and i was told that this is the scalar product...

Hi, this is a rather mathematical question. The inner product between generators of a Lie algebra is commonly defined as \mathrm{Tr}[T^a T^b]=k \delta^{ab} . However, I don't understand why this trace is orthogonal, i.e. why the trace of a multiplication of two different generators is always zero.

Hello everyone, I would like to know if anyone knows what is the inner product for vector fields ##A_\mu## in curved space-time. Is it just: $$ (A_\mu,A_\mu)=\int d^4x A_\mu A^\mu =\int d^4x g^{\mu\nu}A_\mu A_\nu $$ ? Do I need extra factors of the metric? Thanks!

How does one show that eAeB=eA+Be[A,B]/2 where A,B are operators and [ , ] is the commutator. The QM book I am using states it as a fact without proof, but I would like to see how it is proved. I've muddled around with the series expansion, but can't get farther than a few term by term products...

How does one show that eAeB=eA+Be[A,B]/2 where A,B are operators and [ , ] is the commutator. The QM book I am using states it as a fact without proof, but I would like to see how it is proved. I've muddled around with the series expansion, but can't get farther than a few term by term products...

What is the scalar product of orthonormal basis? is it equal to 1 why is a.b=ηαβaαbβ having dissimilar value

Homework Statement Using the equations that are defined in the 'relevant equations' box, show that $$\langle n' | X | n \rangle = \left ( \frac{\hbar}{2m \omega} \right )^{1/2} [ \delta_{n', n+1} (n+1)^{1/2} + \delta_{n',n-1}n^{1/2}]$$ Homework Equations $$\psi_n(x) = \left ( \frac{m...

I'm currently writing my EP on various physical equations including Maxwell's equations, and I had to justify using the dot product of the normal unit vector and the electric field in the integral version. However, I can't think of a reason for not using trigonometry as opposed to the...

Homework Statement Prove the following identity \nabla (\vec{F}\cdot \vec{G}) = (\vec{F}\cdot \nabla)\vec{G} + (\vec{G}\cdot \nabla)\vec{F} + \vec{F} \times (\nabla \times \vec{G}) + \vec{G}\times (\nabla \times \vec{F}) Homework Equations vector triple product \vec{a} \times (\vec{b}...

Show that, for all real $$p$$ and $$m$$, $$e^{2mi\cot^{-1}(p)}\left(\dfrac{pi+1}{pi-1}\right)^m=1$$

Prove $$\cos20^\circ\cdot\cos40^\circ\cdot\cos80^\circ=\frac18$$

Homework Statement Let G = G1 × G2 be the direct product of two simple groups. Prove that every normal subgroup of G is isomorphic to G, G1, G2, or the trivial subgroup. The Attempt at a Solution I tried proving that the normal subgroups would have to be of the form Normal subgroup X Normal...

Say I have a position vector p = e(t) p(t) Where, in 2D, e(t) = (e1(t), e2(t)) and p(t) = (p1(t), p2(t))T And if I conveniently point the FIRST base vector of the frame at the particle, I can use: p(t) = (r1(t), 0)T I want the velocity, so I take v = d(e(t))/dt p(t) + e(t) d(p(t))/dt...

Hello, I hope this is the right forum section. I'm having trouble understanding how calculating the cross product arrives at the final result. When I do something simpler like multiplying a vector by a scalar, I can easily visualize in my head how each component "shrinks" or "grows". With the...

Homework Statement How does the scalar product of displacement four vector with itself give the square of the distance between them? Homework Equations (Δs)2= Δx.Δx ( s∈ distance, x∈ displacement four vector) or how ds2=ηαβdxαdxβ The Attempt at a Solution Clearly I am completely new to the...

When do functions have representations as a "direct product"? For example, If I have a function f(x) given by the ordered pairs: \{(1,6),(2,4),(3,5),(4,2),(5,3),(6,1) \} We could (arbitrarily) declare that integers in certain sets have certain "properties": \{ 1,3\} have property A...

Dear friends !Please help me to solve these two problems.Thanks!

Homework Statement Assume that n > 1 is an integer such that p does not divide n for all primes ≤ n1/3. Show that n is either a prime or the product of two primes. (Hint: assume to the contrary that n contains at least three prime factors. Try to derive a contradiction.) Homework Equations...

Hi, I'm having some issues with a piece of my notes. (relevant pages attached) First we introduce an isomorphism ##U = \oplus_n U_n## from ##\Gamma^{(a)s}\left(\mathcal{H}_1\oplus\mathcal{H}_2\right)## to ##\Gamma^{(a)s}\left(\mathcal{H}_1\right)\otimes\Gamma^{(a)s}\left(\mathcal{H}_2\right)##...

Homework Statement Vectors A and B both have magnitude M. Joined at the tails, they create a 30' angle. What is A x B in terms of M? Homework EquationsThe Attempt at a Solution 0? OR M^2? Sqrt(3)M/3?

So the following question is attached (There is another thread with the same question but no solution to what I am asking on there) Now according to several solutions, apparently IYZ is equal to 0, and they reason this by saying that the geometry is symmetrical. However when looking at the...

Hi their, It's a group theory question .. it's known that ## 10 \otimes 5^* = 45 \oplus 5, ## Make the direct product by components: ##[ (1,1)^{ab}_{1} \oplus (3,2)^{ib}_{1/6} \oplus (3^*,1)^{ij}_{-2/3} ] \otimes [ (1,2)_{ c~-1/2} \oplus (3^*,1)_{ k~1/3} ] = (1,2)^{ab}_{ c~1/2} \oplus...

I'm relatively new to differential geometry and would like to check that this is the correct definition for the tensor product of (for simplicity) two one-forms \alpha,\;\beta\;\;\in V^{\ast} : (\alpha\otimes\beta)(\mathbf{v},\mathbf{w})=\alpha (\mathbf{v})\beta (\mathbf{w}) where...

This is the gravitational potential energy formula $$U = -\int_\infty^r\vec{F}_\text{field}\cdot d\vec{r}$$ If r vector's direction is form infinity to r, then it means it has same direction as Gravitational Force. So cos0=1 But after multiplication there is a negative sign here: "-GMm" $$U =...

Pre-knowledge A matrix is a linear transformation if, T(u+v)= T(u) +T(v) and T(cu)=cT(u). Theorem 8.4.2 If V is a finnite dimensional vector space, and T: V-> V is a linear operator then the following are equivalent. a) T is one to one, b) ker(T)=0, c)...

Let X be an Inner Product Space. If for every closed subspace M, M^{\perp \perp} = M, then X is a Hilbert Space (It's complete). Hint: Use the following map: T : X \longrightarrow \overset{\sim}{X}: T(y)=(x,y)=f(x) where (x,y) is the inner product of X. Relevant equations: S^{\perp} is always...

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...