Product Definition and 1000 Threads

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

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...

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...

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

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...