Product Definition and 1000 Threads

Hi, I'm stuck on a problem from my quantum homework. I have to show <p1|p2> = ∫(from -1 to 1) dx (p1*)(p2) is a scalar product (p1 and p2 are single variable complex polynomials). I've figured out how to show that they satisfy linearity and positive definiteness, but I'm completely stuck on...

Hello! The interior product is defined as ##i_X:\Omega^r(M)\to \Omega^{r-1}(M)##, with X being a vector on the manifold and ##\Omega^r(M)## the vector space of r-form at a point p on the manifold. Now for ##\omega \in \Omega^r(M) ## we have ##i_X\omega(X_1, ... X_{r-1}) = \omega (X,X_1, ...

Hello! I am reading something about differential geometry and I have that for a manifold M and a point ##p \in M## we denote ##\Omega_p^r(M)## the vector space of r-forms at p. Then they say that any ##\omega \in \Omega_p^r(M)## can be expanded in terms of wedge products of one-forms at p i.e...

Hello! (Wave)Let $b_1< b_2< \dots< b_{\phi(m)}$ be the integers between $1$ and $m$ that are relatively prime to $m$ (including 1), and let $B=b_1 b_2 b_3 \cdots b_{\phi(m)}$ be their product. I want to show that either $B \equiv 1 \pmod{m}$ or $B \equiv -1 \pmod{m}$ . Also I want to find a...

I started learning quantum, and I got a bit confused about inner and dot products. I've attached 2 screenshots; 1 from Wikipedia and 1 from an MIT pdf I found online. Wikipedia says that a.Dot(b) when they're complex would be the sum of aibi where b is the complex conjugate. The PDF from MIT...

Homework Statement show that points P, Q and R are in a straight line P (1, -3, 4) Q ( 2, 2, 1) R (3, 7, -2) and find the vectors ## \vec{PQ} ## and ## \vec{QR} ## Homework EquationsThe Attempt at a SolutionIn proving that the points are in a straight line, we might be able to use dot product...

Given the function: $$f(x) = x^3+20x-17.$$ Denote the roots of the function: $r_1,r_2$ and $r_3$. Evaluate (without a calculator) the product: $f(r_1+1) \cdot f(r_2+1) \cdot f(r_3+1)$.

$\tiny{hcc8.11}$ $\textsf{Find product $(1+3i)(2-2i)$}\\$ $8 + 4i$ $\textsf{Then change each to complex form and find product. with DeMoine's Theorem}$ $\textit{ok looked at an example but ??}

Homework Statement If ##A,B,C,D## are positive numbers such that ##A+2B+3C+4D = 8##, then what is the maximum value of ##ABCD##? Homework EquationsThe Attempt at a Solution From AM-GM, I know that the maximum is ##2/3##. However, I want to know exactly what A,B,C, and D must be to take on this...

Hello! I am reading so very introductory stuff on geometric algebra and at a point the author says that, as a rule for calculation geometric products, we have that ##e_{12..n}=e_1\wedge e_2 \wedge ...\wedge e_n = e_1e_2...e_n##, with ##e_i## the orthonormal basis of an n-dimensional space, and I...

Homework Statement Homework Equations The Attempt at a Solution [/B] I could show that the first of the three properties are valid for any value of a,b,c but I couldn’t find a way to show the forth one. Follow all the procedures I already did:

Hi everyone ! I would like to know the real meaning of scalar product. So, I know scalar product is defined as : ||a||.||b||.cos(a;b) = k But what k is ? (Sorry for my english, I am french). Regards :)

So I'm asked to use separation of variables to find a product solution to the given PDE: (5y + 7)du/dx + (4x+3)du/dy = 0 Since it says to find a product solution, I used the form u(x,y) = XY and plugged that into the PDE. However, I am getting stuck because I'm not sure how exactly I should...

Hi! I have an idea for a product that i want to invent / make. I know nothing about materials / science. I came across this website and thought I would post my question. I am looking for a material OR a combo of materials that can withstand the SUN (from the outside) - it will have a lot of...

Is there a spatial derivative that uses the del operator and the double cross product? If so, is it used in physics?

Prove that for $m=2,3,...$ $$\sin\frac{\pi}{m}\sin\frac{2\pi}{m}\sin\frac{3\pi}{m}\cdots\,\sin\frac{(m-1)\pi}{m}=\frac{m}{2^{m-1}}$$

Homework Statement What are the subgroups of Z2 x Z2 x Z2? Homework Equations Hint: There are 16 subgroups. The Attempt at a Solution So far I only manage to get 15 and I am not even sure if these are correct. My answer: $$(0,0,0) , (Z_2,Z_2,Z_2), (1,1,1), (0,0,1), (0,1,0), (1,0,0), (0,1,1)...

Alright, so we ran into a peculiarity in answering this question. Let R be the set of all functions f defined on the interval [0,1] such that - (1) f(t) is nonzero at no more than countably many points t1, t2, . . . (2) Σi = 1 to ∞ f2(ti) < ∞ . Define addition of elements and multiplication...

Hi all, Suppose we have vectors coming in order as A, B and then C (but A must be deleted before C comes in). Then how to get the dot product between A and C? It is allowed to store some calculations of A before deleting elements of A, for example, we could store norm of A, dot(A, B) and etc...

Homework Statement Im given vectors: b = x hat + y hat c = x hat + z hat Homework Equations http://imgur.com/a/iHTOT The Attempt at a Solution so I have 2 eq's... one says: r * s = rscos(theta) the other is a summation saying multiply x component 1 with x component 2, add y component 1...

Homework Statement [/B]Homework Equations a⋅(b x a) = 0 The Attempt at a Solution Is my working below correct? In particular, can you apply the rCB to both the first part of the bracket (0.2j) and the second part (w x rCB) individually like that? My second question relates to: Is that...

I went through an example question that showed me how to solve the question but I'm not sure if I've misunderstood something or if they did a mistake. Question: Derivate y = (1/ax)ax ln(y) = ln( (1/ax)ax ) = ax( ln(1) - ln(ax) ) = -ax ln(ax) (1/y)(dy/dx) = -ax * ax ln(a) - a * ln(ax) dy/dx =...

Hi. I was investigating through this week why there are the differential forms, why are they anti-symmetric, why do we have the Jacobian when expressing the volume in a different coordinate system. This was just fantastic! I found all the connections between these topics. And I found that all of...

Homework Statement I am trying to show that neither ##Z_{p^n}## nor ##\mathbb{Z}## can be written as any family of its proper subgroups. Homework EquationsThe Attempt at a Solution First, I believe this solution (http://www.auburn.edu/~huanghu/math7310/7310-hw2-answer.pdf see problem 6) is...

Why is the dot product equivalent to the matrix multiplication of its components? I've seen some proofs using Pythagorean and cosine law but they don't give you an intuitive feel as to why matrix multiplication works. The geometric definition (##ab cosθ##) is very easy to understand. To a...

It is well known that the product rule for the exterior derivative reads d(a\wedge b)=(da)\wedge b +(-1)^p a\wedge (db),where a is a p-form. In gauge theory we then introduce the exterior covariant derivative D=d+A\wedge. What is then D(a ∧ b) and how do you prove it? I obtain D(a\wedge...

Homework Statement Let ##\{N_i ~|~ i \in I\}## be family of normal subgroups of G such that (i) ##G = \left\langle \bigcup_{i \in I} N_i \right\rangle## (ii) for each ##k \in I##, ##N_k \cap \left\langle \bigcup_{i \neq k} N_i \right\rangle = \{e\}## Then ##G \simeq \prod_{i \in I}^w...

Hello! I was trying to show that the wedge product of 2 one-forms is a 2-form. So we have ## (A \wedge B)_{\mu \nu} = A_\mu B_\nu - A_\nu B_\mu ##. So to show that this is a (0,2) tensor, we need to show that ##(A \wedge B)_{\mu' \nu'} = \Lambda_{\mu'}^\mu \Lambda_{\nu'}^\nu (A \wedge B)_{\mu...

Today, my teacher asked us what is the real life utility of the dot product and cross product of vectors. Many of us said that one gives a scalar product, and one gives a vector product. But he said that, that was not the real life utility of the dot and cross product. He asked us, "Students...

Hi guys, I would like to ask if a set can contain coordinates of points, for example A={[1,3];[4,5];[4,7]} and if we can do Cartesian product of such sets, for example A={[1,3];[4,5]}, B={[7,8];[4,2]} A×B={[1,3][7,8];[1,3][4,2];[4,5][7,8];[4,5][4,2]} (is it correct to write it like that?). I am...

If given a position vector defined for a orthogonal curvilinear coordinate system HOW would the matrices that make up the Levi Civita 3x3x3 matrix remain the same? "Levi Civita 3x3x3 is said to be independent of any coordinate system or metric...

Hi everyone, I am currently working on a subject that involves a lot of 4th order tensors computations including double dot product and inverse of fourth order tensors. First the definitions so that we are on the same page. What I call the double dot product is : $$ (A:B)_{ijkl} =...

Hello I need help to explain the affect of the cross product without the its current symbolism, but for angular momentum. I can explain angular momentum in terms of the cross product of 3D space formulated like this: |r| |v| * sin(angler.v) e-perp to r and v Eq.1 (I can explain this to...

Homework Statement So LG is on 2° position, and you have a hydride shift. So you form one product on 2° and one product on 3°. Which product will be major which will be minor? Homework Equations none. The Attempt at a Solution I know hydride shift will be more major than a methide shift...

Okay so, I am wondering if it is possible to find the actual cross product (not the magnitude of the cross product) from this information 1. magnitude of both vectors 2.angle between vectors 3.plane the vectors lie in Is there any way to calculate that cross product vector? Thank you very much...

Are there practical uses for the formulas for the sum and product of quadratic roots? I have only seen the topic for these sum and product formulas in one section of any college algebra and intermediate algebra books, and then nothing more. I'm just curious if people, ... scientists or...

Homework Statement 5.5 MeV alphas hit Be-9 and produce carbon with a gamma. the Q value is 10.65 I have to find the kinetic energy of the carbon nucleus and the gamma and I am to ignore the gamma ray momentum. Homework Equations T = p^2 / 2*m Q = T_be + T_alpha - T_c - T_gam[/B]The Attempt...

In reviewing some calculations, I've arrived at the series: ##S(d)=-\frac 1 {d-1}+\frac 1 2 \frac{d-2}{d-3}-\frac 1 8 \frac{(d-2)(d-4)}{d-5}+\frac 1 {48} \frac{(d-2)(d-4)(d-6)}{d-7}+\dots ## Its an infinite series but because I'm interested in its values for even ##d##s, its actually a finite...

It is very well known result that ##grad[e^{i\vec{k}\cdot \vec{r}}]=i\vec{k}e^{i\vec{k}\cdot \vec{r}}##. Also ##\vec{k}\cdot \vec{r}=kr\cos \theta## and ##gradf(r)=\frac{df}{dr} grad r##. Then I can write grad e^{ikr\cos \theta}=ik\cos \theta e^{i \vec{k}\cdot \vec{r}}...

In a beam splitter with one vacuum input, the output looks like ##\frac{1}{\sqrt 2}(|1_A\rangle +i|0_B \rangle)\frac{1}{\sqrt 2}(-i|1_A\rangle +|0_B \rangle)##. If there is some further processing, the vacuum ##i|0_B\rangle##, along with the i , could end up multiplied with some non-vacuum term...

Homework Statement Predict the major product and type of reaction mechanism (E1/E2/SN1/SN2) Homework EquationsThe Attempt at a Solution Could someone just double check my work for this? From top to bottom I got: E2, E1, SN1, E1

Suppose a second rank tensor ##T_{ij}## is given. Can we always express it as the tensor product of two vectors, i.e., ##T_{ij}=A_{i}B_{j}## ? If so, then I have a few more questions: 1. Are those two vectors ##A_i## and ##B_j## unique? 2. How to find out ##A_i## and ##B_j## 3. As ##A_i## and...

As I am also attaching solution along with the problem there is no point in posting this thread in homework forums . In the following question (solution is also there) https://s29.postimg.org/omlnt73lz/IMG_20170407_090748.jpg https://s18.postimg.org/hsuxm5uvt/IMG_20170407_090805.jpg why are we...

The inner product axioms are the following: ##\text{(a)} \ \langle x+z,y \rangle = \langle x,y \rangle + \langle z,y \rangle## ##\text{(b)} \ \langle cx,y \rangle = c\langle x,y \rangle## ##\text{(c)} \ \overline{\langle x,y \rangle} = \langle y,x \rangle## ##\text{(d)} \ \langle x,x \rangle > 0...

Homework Statement Let ##V## be a vector space equipped with an inner product ##\langle \cdot, \cdot \rangle##. If ##\langle x,y \rangle = \langle x, z\rangle## for all ##x \in V##, then ##y=z##. Homework EquationsThe Attempt at a Solution Here is my attempt. ##\langle x,y \rangle = \langle x...

Hi, hopefully a quick question here...how do you calculate the angle between two vectors if the only information you have is the value of their scalar product and the magnitude of their cross product? Thanks! Andy

Homework Statement Let ##A## and ##B## be finite groups, and ##A \times B## be their direct product. Given that ##(a,1)## and ##(1,b)## commute, and that ##(a,1)^n = (a^n,1)## and ##(1,b)^n = (1,b^n)## for all a and b, show that the order of ##(a,b)## is the least common multiple of the orders...

Hello, I have a question about why I can't determine the angle between two vectors using their cross product. Say there are two vectors in the XY-plane that we want to find the angle between: A = -2.00i + 6.00j B = 2.00i - 3.00j The method to do this would be to work out the scalar product of...

Show that the product of the roots of the equation x^2 + px + q = 0 is q. I need help with the set up. Must I use the discriminant here?

In group theory, what is the direct product of a symmetry group with itself? Say T*T or O*O?