Product Definition and 1000 Threads

Hi, I am working through a textbook on general relativity and have come across the statement: "A general (2 0) tensor K, in n dimensions, cannot be written as a direct product of two vectors, A and B, but can be expressed as a sum of many direct products." Can someone explain to me how this...

Hi, I am interested to see how to use the vector dot product formula in spherical coordinate system, $$ V_1= r + \theta, at (1,0,0)$$ and $$ V_2= r - \theta, at (1, \frac{\pi}{2}, \frac{\pi}{2})$$ how to evaluate their dot product? do I have to transfer them into cartesian system? what would...

Homework Statement If ##u,v,w\in\mathbb{R}^3##, show that ## u\times(v\times w) = (u.w) v - (u.v) w ##. Homework Equations The Attempt at a Solution Since ## u\times(v\times w)##, ##v## and ##w## are orthogonal to ##v\times w##, these vectors are coplanar. Therefore, there must be reals ##...

Why wouldn't this work?

Homework Statement Show that the product of two nxn unitary matrices is unitary. Is the same true of the sum of two nxn unitary matrices? Homework Equations Unitary if A†A=I Where † = hermitian conjugate I = identity matrix. The Attempt at a Solution [/B] We have the condition: (AB)†(AB)=I I...

Reading a book about 3d math, and I am confused as to what happened on this Vector Cross Product problem. I'm thinking there was just an error that wasn't caught. For the first row, instead of (3)(8)-(-4)(-5) shouldn't it have been (3)(8)-(4)(-5) and had the same displayed result of 44? And for...

Homework Statement Refer to solution II , the author used the scalar analysis( dot product) to get the direction of moment ...IMO , this is incorrect ... Only cross product can be determined this way . correct me if I'm wrong . Homework EquationsThe Attempt at a Solution

Problem. Let $R$ be a local ring (commutative with identity) ans $M$ and $N$ be finitely generated $R$-modules. If $M\otimes_R N=0$, then $M=0$ or $N=0$. The problem clearly seems to be an application of the Nakayama lemma. If we can show that $M=\mathfrak mM$ or $N=\mathfrak mN$, where...

Homework Statement Hi, I wasn't sure whether to post this here or in the engineering forums. Since it's mainly math/theory I figured here would be more appropriate. Feel free to move it if it doesn't belong here. All relevant info etc. is in the picture, thanks. Homework Equations The...

Homework Statement Given basis |x>,|y>,|z> such that <x|x> = 2,<y|y> = 2,<z|z> = 3,<x|y> = i, <x|z> = i, and <y|z> = 2. Build an orthonormal basis|x'>,|y'>,|z'>. Each of the new basis vectors should be expressed in terms of the old ones multiplied by coefficients. Homework Equations |x'> =...

Homework Statement The diagram shows a box with parallel faces. Two of the faces are trapezoids and four of the faces are rectangles. The vectors A, B, and C lie along the edges as shown, and their magnitudes are the lengths of the edges. Define the necessary additional symbols and prove...

Here is a mystery I'm trying to understand. Let ##\hat{U}(t) = \exp[-i\hat{H}t]## is an evolution operator (propagator) in atomic units (\hbar=1). I think I'm not crazy assuming that ##\hat{U}(-t)\hat{U}(t)=\hat{I}## (unit operator). Then I would think that the following should hold \left\langle...

Homework Statement [/B] Vector A lies in the yz plane 63.0 degrees from the +y axis, has a positive z component, and has a magnitude 3.20 units. Vector B lies in the xz 48.0 degrees from the +x axis, has positive z component, and has magnitude 1.40 units. a) find A dot B b) find A x B c)...

Hello, PF! I had a quick question that I hoped maybe some of you could help me answer. The question is simple: Why is the cross product of two parallel vectors equal to the zero vector? I can see this easily mathematically through completing the cross product formula with two parallel...

The determinant of a 3x3 matrix can be interpreted as the volume of a parallellepiped made up by the column vectors (well, could also be the row vectors but here I am using the columns), which is also the scalar triple product. I want to show that: ##det A \overset{!}{=} a_1 \cdot (a_2 \times...

Just a question : do we have in Dirac notation $$\langle u|A|u\rangle\langle u|B|u\rangle=\langle u|\langle u|A\otimes B|u\rangle |u\rangle$$ ?

Homework Statement I'm supposed to prove det A = \frac{1}{6} \epsilon_{ijk} \epsilon_{pqr} A_{ip} A_{jq} A_{kr} using the triple scalar product. Homework Equations \frac{1}{6} \epsilon_{ijk} \epsilon_{pqr} A_{ip} A_{jq} A_{ kr} (\vec u \times \vec v) \cdot \vec w = u_i v_j w_k...

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...