Product Definition and 1000 Threads

Homework Statement Find the gradient of \underline{\nabla}(\underline{a}\cdot\underline{r})^n where a is a constant vector, using suffix notation and chain rule. Homework Equations On the previous problem,s I found that grad(a.r)=a and grad(r)=\underline{\hat{r}} The Attempt at a Solution...

I would like to gain a more formal mathematical understanding of a construct relating to spinors. When I write down Dirac spinors in the Weyl basis, I see why if I multiply the adjoint (conjugate transpose) of a spinor with the original spinor I don't get a SL(2,C) scalar. It just doesn't work...

Prove $$\sum_{n=0}^N\cos(nx)=\csc\left(\dfrac x2\right)\sin\left(\dfrac{(N+1)x}{2}\right)\cos\left(\dfrac{Nx}{2}\right)$$ I've tried working from the RHS with various identities but haven't managed to come up with anything that works. I suspect this problem involves some trigonometry that I...

Homework Statement Homework Equations The product rule formula. The Attempt at a Solution I managed to solve 45/50 product rule but I can't seem to solve these ones. Apparently you use product rule to solve these.

Let's say I have two particles A and B and I want to find the total charge parity of the system ##C_{AB}##. In what cases is it allowed to say ##C_{AB}=C_{A}.C_{B}##? I suspect that if A and B are their own antiparticles, then that is OK. Is this even the case when the system has a relative...

Homework Statement Explain in your own words why the product of eigenvalues of any diagonalisable N × N matrix A must equal the determinant of A. Homework Equations MT=M-1 The Attempt at a Solution So what I do know: the determinant measures the change in area of the unit square under the...

Homework Statement Basically, I'm looking at the property that says if the magnitude of a vector valued function is constant, then the vector function dotted with it's derivative will be zero. But I'm stuck towards the end because the proof I found online seems to skip a step that I'm not...

Homework Statement Suppose R and Q are two quantum systems with the same Hilbert space. Let |i_R \rangle and |i_Q\rangle be orthonormal basis sets for R and Q . Let A be an operator on R and B an operator on Q . Define |m\rangle := \sum_i |i_R\rangle |i_Q\rangle ...

I'm not 100% confident of my approach to the 2 exercises below: Orbital angular momentum of i'th element is $\vec{L_i} = \vec{r_i} \times \vec{p_i} = m_i \vec{r_i} \times (\omega \times \vec{r_i}) $ a) Find the inertia matrix $I$ such that (omitting vector signs from here on) $L = I \omega, |L...

At the risk of sounding ignorant I'd like to propose a question to someone well versed in Homological Algebra and General Relativity. I'm starting to study the tensor product functor in the context of category theory because I'm interested in possibly doing a paper on TQFT for a directed...

If the cross product in ℝ3 is defined as the area of the parallelogram determined by the constituent vectors joined at the tail, how does one go about proving this product to distribute over vector addition? I've attached a drawing showing cyan x yellow, cyan x magenta, and cyan x (magenta +...

Can someone please help me prove this product rule? I'm not accustomed to seeing the del operator used on a dot product. My understanding tells me that a dot product produces a scalar and I'm tempted to evaluate the left hand side as scalar 0 but the rule says it yields a vector. I'm very confused

Here is a simple question : let f(g(x)) = h(x)*g(x). I want to calculate df/dx. If I use the product rule, I get g(x)h'(x) + h(x)g'x). Now if I use the composition/chain rule, I get df/dx = df/dg * dg/dx = h(x) * g'(x) which is different. I guess my df/dg = h is wrong, but I can't see what...

Hi there, I understand that taking the dot product of two four vectors automatically applies the metric tensor to the second vector. Is there a way to take write the dot product, using vector notation, in a way which keeps the signs of all of the components positive? Thanks in advance.

Question: The following are determinants of partial derivatives multiplied together giving another determinant of partial derivatives Prove that this equality holds: Relevant Equations: |du/dx du/dy| |dx/dr dx/ds| |du/dr du/ds| |dv/dx dv/dy| |dy/dr dy/ds| = |dv/dr dv/ds| Attempt at Solution: I...

On pages 67 & 68 of Hassani's mathematical physics book, he gives the following definition: "Let ## \mathcal{A} ## and ## \mathcal{B} ## be algebras. The the vector space tensor product ## \mathcal{A} \otimes \mathcal{B} ## becomes an algebra tensor product if we define the product ##...

Homework Statement Let A be an nxn matrix, and let |v>, |w> ∈ℂ. Prove that (A|v>)*|w> = |v>*(A†|w>) † = hermitian conjugate Homework EquationsThe Attempt at a Solution Struggling to start this one. I'm sure this one is likely relatively quick and painless, but I need to identify the trick...

I'm trying to understand how the derivative of this function: x=ρcosθ Becomes this: dx=−ρsinθdθ+cosθdρ First off I'm guessing that x is a function of both ρ AND cosθ, or else we wouldn't be using the product rule in the first place..Am I correct? So how could we write this in functional...

4Use the power to sum formula to simplify the expression $\frac{\sin\left({3\theta}\right)+\sin\left({5\theta}\right)} {\cos\left({3\theta}\right)+\cos\left({5\theta}\right)}$ The answer is $\tan(4\theta)$ $$\sin\left({3\theta}\right)+\sin\left({5\theta}\right)...

Hi - from orbital angular momentum components, $[L_x, L_y] = iL_z$ My book claims 'Hence, $ \vec{L} \times \vec{L} = i\vec{L} $' I'm keen to know how they get that, an also why that cross products isn't = 0, like $A \times A$ would be ?

I got to here in a simple exercise (orb. ang. momentum cords), realized I was applying something I didn't understand ... $L = -i \begin{vmatrix}\hat{x}&\hat{y}&\hat{z}\\x&y&z\\\pd{}{x}&\pd{}{y}&\pd{}{z}\end{vmatrix}$ I 'know' it equates to $L_x =-i \left( y\pd{}{z} - z\pd{}{y} \right) $ - but...

I know the bac-cab rule, but add $\nabla$ and it's not so clear .. applying it to $\nabla \times \left( A \times B \right) = A\left(\nabla \cdot B\right) - B\left(\nabla \cdot A\right) ...$, not quite Please walk me through why the other 2 terms emerge ?

Homework Statement at what initial speed would a projectile have to start at when ejected at 35 degrees to the horizontal from a point A to a point B which is 9.4km distance away in the horizontal and 3.3km below it. taking g as 10m/s[/B]Homework Equations I'm not really sure if these equations...

Homework Statement Angular momentum is the cross product of r and mv. But why is there mvR outside of the paranthesis? And where did the v go in the second paranthesis - shouldn't the second paranthesis be (-v*sin(ωt), v* cos(ωt)). Does anyone have any idea how they did the cross product...

Let $a,\,b,\,c$ be real numbers greater than $2$ such that $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=1$. Prove that $(a-2)(b-2)(c-2)\le 1$.

Homework Statement Consider a qubit in the state |v> ∈ ℂ^2. Suppose that a measurement of δn is made on the qubit. Show that the probability of obtaining the result "+1" in the measurement is equal to 0 if and only if |v> and |n,+> are orthogonal. Homework Equations Inner product axioms |v>|w>...

Homework Statement Homework EquationsThe Attempt at a Solution I was able to do the second part of part a using integration by parts. But I am having no luck for the first part, proving that the inner product is positive definite. Pointers are appreciated!

Hello, As part of a project I'm working on, I need to mount a shaft to a sheet of plywood or plastic. On this shaft will sit a gear which rotates. The shaft, however, should not rotate or move in any way. I have tried using a threaded shaft, mounting it to wood just by screwing in two nuts on...

Lets look at the force on a wire segment in a uniform magnetic field F = I∫(dl×B) I am curious if, from this, we can say: F = I [ (∫dl) × B] since B is constant in magnitude and direction

1)I use Linux Mint 17.2 and wxMaxima 13.04.2. In wxMaxima 13.04.2, the code below, It plays correctly: plot2d([x,x^3,[discrete,[[0,0],[1,1],[-1,-1]]]],[x,-5,5],[y,-5,5], [style,[lines,2,1],[lines,2,4],[points,3,2]],[point_type,bullet], [legend,"x","x^2",""],[xlabel, "x"], [ylabel...

A method for finding the shortest distance between 2 skew, non intersecting lines is to 1st find the common normal, using $ \vec{n} = \frac{\vec{v_1} \times \vec{v_2}}{|\vec{v_1} \times \vec{v_2}|} $ I'm looking for a proof or intuition as to why this is true please? Then apparently we get the...

I am reading Paul E. Bland's book: Rings and Their Modules and am currently focused on Section 2.1 Direct Products and Direct Sums ... ... I am trying to fully understand Bland's definition of a direct product ... and to understand the motivation for the definition ... and the implications of...

Homework Statement Two vectors A and B have magnitude A = 3.00 and B = 3.00. Their vector product is A x B= -5.00k + 2.00i. What is the angle between A and B? Homework Equations Magnitude of vector product = magnitude of A * magnitude of B * sin of the smaller angle between A and B...

Homework Statement I'm having trouble understanding the definition of a complex inner product. Let λ ∈ ℂ So if we have <λv|w> what does it equal to? Does it equal λ*<v|w> where * is the complex conjugate?Are all these correct: <λv|w> = λ*<v|w> <v|λw> = λ<v|w> <v|w> = (<w|v>)* <v|w> = Σvw...

Homework Statement Find a direct product representation for the quaternion group. Which are your options? Homework EquationsThe Attempt at a Solution Theorem: The internal direct product of normal subgroups forms a homomorphism of the group...

1. Problem statement: Assume that u is a vector and A is a 2nd-order tensor. Derive a transformation rule for a 3rd order tensor Zijk such that the relation ui = ZijkAjk remains valid after a coordinate rotation.Homework Equations : [/B] Transformation rule for 3rd order tensors: Z'ijk =...

Why work done by a force was taken as dot product between force applied and displacement caused?

Homework Statement S = a non-empty set of vecotrs in V S' = set of all vectors in V which are orthogonal to every vector in S Show S' = subspace of V Homework Equations Subspace requirements. 1. 0 vector is there 2. Closure under addition 3. Closure under scalar multiplication The Attempt at...

Homework Statement Prove that <v|0>=0 for all |v> ∈ V. Homework EquationsThe Attempt at a Solution This is a general inner product space. I break it up into 2 cases. Case 1: If |v> = 0, the proof is trivial due to inner space axiom stating <0|0> = 0. Case 2: If |v> =/= 0 then: I use <v|0>...

Hi, I was just wondering if you have a cross product can you multiply out the constants and put them to one side. So ik x ik x E is equal to i^2(k x k x E) therefore is equal to -k x k x E. Is that correct?

Im stuck on theorem 5 where the book used chain rule then used product rule then again using the chain rule. How in the world does it work? I don't get product rule used and chain rule used after.

Homework Statement Good day, I need to show that S_n=\mathbb{Z}_2(semi direct product)Alt(n) Where S_n is the symmetric group and Alt(n) is the alternating group (group of even permutations) note: I do not know the latex code for semi direct product Homework Equations none The Attempt at...

Hey it might be a stupid question but I saw that the tensor product of 2 vectors with dim m and n gives another vector with dimension mn and in another context I saw that the tensor product of vector gives a metrix. For example from sean carroll's book: "If T is a (k,l) tensor and S is a (m, n)...

So yeah, I understand that you can calculate torque as F*d, and you get a "number". But when you calculate a cross product of torque, r x F, what does that actually give you? It is a vector, perpendicular to F and r, but what "is" that? I mean, is it like an axis around which the object is...

Homework Statement Good day all! (p.s I don't know why every time I type latex [ tex ] ... [ / tex ] a new line is started..sorry for this being so "spread" out) So I was wondering if my understanding of this is correct: The Question asks: "\mathbb{Z}_4 has a subgroup is isomorphic to...

Homework Statement The problem is given in the following photo: Actually I did the first proof but I couldn't get the second relation. (Divergence of E cross H). Homework Equations They are all given in the photo. (a) (b) and (c). The Attempt at a Solution What I tried is to interchange...

Homework Statement [/B] If 1.20 grams of salicylic acid is reacted with excess methanol, what mass of ester should you expect to achieve theoretically? Molar masses of.. salicylic acid is 138.13g/mol methanol is 32.05g/mol ester (methyl salicylate) is 152.16g/mol Mole ratio...

Hey! :o I am looking at the following exercise: Consider the ellipse $$\frac{x^2}{p^2}+\frac{y^2}{q^2}=1$$ where $p > q > 0$. The eccentricity of the ellipse is $\epsilon =\sqrt{1-\frac{q^2}{p^2}}$ and the points $(\pm \epsilon p, 0)$ on the $x$-axis are called the foci of the ellipse, which...

So basicly our teacher taught us in high school how to find the product of some equations but I do not understand it very well and I need someone to teach me how to solve this basic problem. The Equation is : (3x^2-4x+1)(4x^2+x-2) I do not know how to find the product of that problem can...

Homework Statement Figure out the minimum sum of products for g(r s t) = r't' + rs' +rs 2. The attempt at a solution I understand you can simplify it with the Boolean theorems (e.g r't' + r = t' + r) , however how would you solve it using K-maps? I drew out a truth table, but it seems as if...