Product Definition and 1000 Threads

Hi. If I have a vector v , say for velocity for example then v.v = v2 and I differentiate wrt t v.v I get 2v.dv/dt but if I differentiate v2 I get 2v dv/dt but v.dv.dt is not the same as v dv/dt so what am I doing wrong ? Thanks

i am learning physics for fun and also to make taking the actual courses easier. i have a physics textbook because i took physics for my associate's degree but ended up withdrawing because i took too many courses. anyway, i like "connected" knowledge in math though i know it is a bit of a...

Homework Statement Is the following conclusion correct? Assume there's an equation with vectors on both sides. Taking the dot product of this equation with vectors on both sides loses information, but information will not lose when taking dot products with higher rank tensors on both sides...

How may one show that, $$\int_{0}^{4\pi}\sin\left({x\over 2}\right)\sin\left({x\over 4}\right)\ln^2\left[\sin\left({x\over 8}\right)\right]\mathrm dx={\pi\over 9}[12\ln (2)-1]$$

I read from this page https://properphysics.wordpress.com/2014/06/09/a-no-nonsense-introduction-to-special-relativity-part-6/ that the basis vectors are the canonical basis vectors in any coordinate system. This seems to be wrong, because if that was the case the metric would be the identity...

Homework Statement a: In plane polar coordinates, find the scalar product of the vector (0,1) with itself. b: What would be the r, θ components of the unit vector in the θ direction? Homework Equations Scalar product of 2 vectors = AαgαβBβ The Attempt at a Solution For part a, I used the...

Hello, I am currently stuck on problem 5.3 (c) about spinor products in PS, where one needs to prove the Fierz identity: $$ \bar{u}_{L}(p_{1}) \gamma^{\mu} {u}_{L}(p_{2}) [\gamma_{\mu}]_{ab} = 2 [u_{L}(p_{2})\bar{u}_{L}(p_1) +u_{R}(p_{1})\bar{u}_{R}(p_2) ]_{ab} $$ They say that a Dirac matric M...

Prove, that $$\prod_{j = 1}^{n}\left(1+2\cos \left(\frac{3^j}{3^n+1}2\pi\right)\right) = 1.$$

Homework Statement The product of two positive numbers is 100. What numbers will produce the least possible sum? Confirm that the sum is in fact a minimum. Homework EquationsThe Attempt at a Solution For this question here I feel like the wording is a bit confusing, I tried my best please let...

Consider that a vector can be represented in two different basis. My question is do we need to normalize both basis before taking the inner product? What motivates this question is because I found out that the inner product of a vector having components ##a,b## in the normalized polar basis of...

Homework Statement In a physics problem where V is the volume i have ##\displaystyle~3V-\frac{3}{4}V~##. i get 2 different answers when i calculate. Homework Equations $$a(b-c)=ab-ac$$ The Attempt at a Solution I can: $$3V-\frac{3}{4}V=3\left( 1-\frac{1}{4} \right)V=3\frac{3}{4}V$$ And if i...

Is there a reaction of which the product is the catalyst for that same reaction?

Homework Statement Five consecutive multiples of 8 have a 9-digit product of ##49xyz2160##. What is the value of ##x\cdot y \cdot z##? Homework Equations I am unsure of what equations would be relevant. The Attempt at a Solution I tried breaking the number into its parts: ##4\cdot 10^8+9\cdot...

Reading the Wikipedia entry about the Navier–Stokes equation, and I don't understand this second term, the one with the outer product of the flow velocities. I mean, I understand the literal mathematical meaning, but I don't have an intuitive idea of what it physically represents. When I make...

EDIT -- "Who wants it" -- Obviously with no obligation to me, since I'm publishing it on a public forum. :smile: So I just saw the very cute "Human Kindness" commercial on TV again, where the Dad sees his son with respiratory issues having trouble blowing out the candle on his 1 y/o birthday...

Homework Statement Homework Equations I am not sure. I have not seen the triangle inequality for inner products, nor the Cauchy-Schwarz Inequality for the inner product. The only thing that my lecture notes and textbook show is the axioms for general inner products, the definition of norm...

Hello Are there any methodologies to assess whether a manufacturer should invest to manufacture and sell a new product? I am looking for any Frameworks or procedures that could help methodically and systematically assess an investment like this.

Hello, I am having trouble understanding a proof presented here: http://www.fen.bilkent.edu.tr/~ercelebi/Ax(BxC).pdf This is a proof of the triple product identity, but I don't understand the last step, where they calculate ##\lambda##. Don't you lose all generality when you state ##\vec A##...

Homework Statement Taking derivative of 3 term product. ## \frac{d}{dx} (3x^3 y^2 y'^2) ## Homework Equations I read that (abc)' = (ab)c' + (bc)a' + (ca)b' The Attempt at a Solution ## 9x^2(y^2y'^2) + 6x^3yy'^2 + 6x^3y^2y' ## is this correct ?

1) I have a list of customers who have bought a product and consumed it on a certain date. I receive every month an excel sheet with the following columns: Customer_Account / Activation_Date / Country / 2) I have another excel sheet with the volume of sales of that product by Country. I also...

Evaluate without the use of a calculator the product:$P = \sin 1\sin 2\sin 3…\sin 89$ (all angles in degrees)

Homework Statement Show that the sequence of partial sums s_{n} = 1+\sum_{i=1}^{n} \left(\prod_{k=1}^{i}\left( \frac{1}{2} + \frac{1}{k}\right)\right) converges, with n\in \mathbb{N}\cup \{0\} Homework EquationsThe Attempt at a Solution [/B] So we want to find \lim_{n\to\infty} s_{n} =...

Homework Statement [/B] Polynomial P(x) when divided by (x-2) gives a remainder of 10. Same polynomial when divided by (x+3) gives a remainder of 5. Find the remainder the polynomial gives when divided by (x-2)(x+3). 2. Homework Equations Polynomial division, remainder theorem The Attempt...

Is there some easy way to see that [\vec{p}^2, \vec{p} \times \vec{L}] is equal zero? I use component method and got that.

Homework Statement Prove that ##\vec {a} \cdot (\vec {b} \wedge \vec {C_r}) = \vec {a} \cdot \vec {b} \vec {C_r} - \vec {b} \wedge (\vec {a} \cdot \vec {C_r})##. Note that ##\vec {a}## is a vector, ##\vec {b}## is a vector, and ##\vec {C_r}## is an r-blade with ##r > 0##. Also, the dot...

I am trying to follow modern QFT by Tom Banks and I am having an issue with literally the first equation. He claims that beginning from ## |p_1 , p_2, ... , p_k> \: = \: a^\dagger (p_1) a^\dagger (p_2) \cdots a^\dagger (p_k)|0> ## with the commutation relation ##[a (p),a^\dagger (q)]_\pm \: =...

Suppose f,g:ℂ→ℂ are analytic with singularities at z=0. I was wondering whether f(z)^2 or f(z)g(z) will have a singularity at z=0? For each, can you give me a proof or a counterexample?

My question is simply whether you can reduce a vector triple product, or more generally a scalar multiplier of a vector in a cross product? Given: (A x (uB x C) = v, where u and v are known constants. Is it valid to change that to: u(A x (B x C) = v or (A x uB) = v, can you change that to u(A...

Given non-negative reals, $\alpha_i$, where $i = 1,2,...,n.$ Prove, that $\alpha_1+\alpha_2+...+\alpha_n \leq \frac{1}{2}$ $\Rightarrow$ $(1-\alpha_1)(1-\alpha_2)...(1-\alpha_n) \geq \frac{1}{2}.$

I'm reading up on the Lagrangian equation, but what I'm asking is to do with electromagnetism. In the first equation here: http://www.phys.ufl.edu/~pjh/teaching/phy4605/notes/chargelagrangiannotes.pdf L equals the kinetic minus the potential energy. For the potential energy term, I just don't...

I have a very basic knowledge of calculus of one variable . In the chapter on heat and thermodynamics , ideal gas law PV =nRT is given . Then the book says, differentiating you get PdV +VdP = nRdT . The book doesn't explain the differentiation step . I think , there are two ways to...

Homework Statement question concerning part c. Homework Equations The question is pretty simple if there is no zero of order ##N## at infinity, such that it does not cancel the pole of ##f(t)## at infinity of order ##N##. In this case it follows that ## f(t) g(t) \in M^{!}_2 ## and so we...

Homework Statement Given two sets of Cartesian product S=A1×A2...×An P=(A1×A2...×An-1)×An show that there exists bijection between the two sets. Homework Equations ∀a1,a2:a1∈A1, a2∈A2: A1×A2=(a1,a2) The Attempt at a Solution let f be a function that maps f: P → A1×A2...×An-1 where...

Hi, what is the physical meaning, or also the geometrical meaning of the inner product of two eigenvectors of a matrix? I learned from the previous topics that a vectors space is NOT Hilbert space, however an inner product forms a Hilbert space if it is complete. Can two eigenvectors which...

Hi, I am trying to prove that the eigevalues, elements, eigenfunctions or/and eigenvectors of a matrix A form a Hilbert space. Can one apply the inner product formula : \begin{equation} \int x(t)\overline y(t) dt \end{equation} on the x and y coordinates of the eigenvectors [x_1,y_1] and...

Show that for integer a,b,c the product $abc(a^3-b^3)(b^3-c^3)(c^3-a^3)$ is divisible by 7

I am trying to figure how to get 1. from 2. and vice versa where the e's are bases for the vector space and θ's are bases for the dual vector space. 1. T = Tμνσρ(eμ ⊗ eν ⊗ θσ ⊗ θρ) 2. Tμνσρ = T(θμ,θν,eσ,eρ) My attempt is as follows: 2. into 1. gives T = T(θμ,θν,eσ,eρ)(eμ ⊗ eν ⊗ θσ ⊗ θρ)...

Homework Statement Show: ##D^{(n-m)} (x^2-1)^n = \frac{(n-m)!}{(n+m)!} (x^2-1)^m D^{(n+m)} (x^2-1)^n## Hint: ##D^{(n-m)} (x^2-1)^n = D^{(n-m)} [(x-1)^n (x+1)^n]## Homework Equations [/B] Leibniz Rule for Differentiation: $$D^k (uv) = \sum_{j=0}^k \binom{k}{j} D^j (u) D^{(k-j)} (v)$$ The...

Hi everybody, I'm writing some algebra classes in C++ , Now I'm implementing the modified sparse row matrix , I wrote all most all of the class, but I didn't find the way saving computing time to perform the product of two Modified sparse row matrix .. if you don't know it you can read in the...

I'm reading a textbook on electromagnetism. It says that for two vector fields ##\textbf{F}(\textbf{r})## and ##\textbf{G}(\textbf{r})## their inner product is defined as ##(\textbf{F},\textbf{G}) = \int \textbf{F}^{*}\cdot \textbf{G} \thinspace d^3\textbf{r}## And that if ##\textbf{F}## is...

Homework Statement Let ##\{G_i \mid i \in I\}## be a family of groups, then ##\prod^w G_i##, the external weak direct product, is the internal weak direct product of the subgroups ##\{i_k(G_k) \mid k \in I\}##, where ##i_k : G_k \to \prod G_i## is the canonical embedding. Homework...

Homework Statement Let ##H, K, N## be nontrivial normal subgroups of a group ##G## and suppose that ##G = H \times K##. Prove that ##N## is in the center of ##G## or ##N## intersects one of ##H,K## nontrivially Homework EquationsThe Attempt at a Solution I presume that ##G = H \times K##...

Hello! Kunneth fromula states that for 3 manifolds such that ##M=M_1 \times M_2## we have ##H^r(M)=\oplus_{p+q=r}[H^p(M_1)\otimes H^q(M_2)]##. Can someone explain to me how does the tensor product acts here? I am a bit confused of the fact that we work with r-forms, which are by construction...

Hello! The cohomology ring on an M-dim manifold is defined as ##H^*(M)=\oplus_{r=1}^mH^r(M)## and the product on ##H^*## is provided by the wedge product between cohomology classes i.e. ## [a]## ##\wedge## ##[c]## ##= [a \wedge c]##, where ##[a]\in H^r(M)##, ##[c]\in H^p(M)## and ##[a \wedge...

Homework Statement What will result in the reaction of benzene with acetone in sulfuric acid? Homework Equations The Attempt at a Solution Is bisphenyl correct?

I am new to quantum mechanics and I have recently been reading Shankar's book. It was all good until I reached the idea of representing functions of continouis variable as kets for example |f(x)>. The book just scraped off the definition of inner product in the discrete space case and refined it...

So, while textbook was deriving Liouville eq, this is one of step the book uses. I don't understand this step at all. Why is this step true?

Hello. I was trying to prove that the tangent bundle TM is a smooth manifold with a differentiable structure and I wanted to do it in a different way than the one used by my professor. I used that TM=M x TpM. So, the question is: Can the tangent bundle TM be considered as the product manifold...

Greetings, can somebody show me how to calculate such a term? P= X E² where X is a third order tensor and E and P are 3 dimensional vectors. Since the result is supposed to be a vector, the square over E is not meant to be the scalar product. But the tensor product of E with itself yields a...

Hi, In a demostration i found a change in order i can't understand. how can the differential pass in the beginning part? the only thing I'm sure is that "v" and "dr" are parallel