Product Definition and 1000 Threads

Hello! (Wave) There is the following sentence in my notes: Let $A$ be a set. We define the set $I_A=\{ <a,a>, a \in A \}$. $$A \times A=\{ <a_1,a_2>: a_1 \in A \wedge a_2 \in A \}$$ Then $I_A$ is a relation, but does not come from a cartesian product of sets. Could you explain me the last...

Homework Statement [/B] Let Z be any 3×3 orthogonal matrix and let A = Z-1DZ where D is a diagonal matrix with positive integers along its diagonal. Show that the product <x, y> A = x · Ay is an inner product for R3. Homework Equations None The Attempt at a Solution I've shown that x · Dy is...

Homework Statement Find angle between vectors if \cos\alpha=-\frac{\sqrt{3}}{2} [/B]Homework EquationsThe Attempt at a Solution Because cosine is negative I think that \alpha=\frac{5\pi}{6}. But also it could be angle \alpha=\frac{7\pi}{6}. Right? When I search angle between vectors I do not...

Hi, Let us suppose we have three real matrices A, B, C and let \circ denote the Hadamard product, while AB is the conventional matrix product. Is this relation true for all A, B, C matrices: C \circ (AB) = A( C\circ B)? I looked at it more thoroughly and I realized that this assumption is...

If I want to take the wedge product of $$\alpha = a_i\theta^i $$ and $$\beta = b_j\theta^j$$ I get after applying antisymmetrization,$$ \alpha \Lambda \beta = \frac{1}{2}(a_ib_j - a_jb_i)\theta^i\theta^j$$ My question is it seems to me that antisymmetrization technique doesn't apply to the...

Homework Statement Considera 1.0 C charge moving with a velocity of v = -2.0i + 2.0j - 1.0k in a magnetic field of B = -4.0i + 1.0j – 3.0k. What force is this charge experiencing? What is the angle between the velocity and magnetic field vectors? Homework Equations F = q(E + v x B)...

Hi! (Wave) If $A,B$ are sets, the set $\{ <a,b>=\{ a \in A \wedge b \in B \}$ is called Cartesian product of $A,B$ and is symbolized $A \times B$. If $A,B,C$ sets, then we define the Cartesian product of $A,B,C$ as: $$A \times B \times C:=(A \times B) \times C$$ But.. is it: $(A \times B)...

Hi all, Went to a seminar today, arrived a few minutes late; hope someone can tell me something about this topic and/or give a ref so that I can read on it . I know this is a lot of material; if you can refer me to at least some if, I would appreciate it : 1)Basically, understanding how/why the...

Homework Statement Use the LC symbol to calculate the following: $$\nabla \times \frac{\vec{m} \times \hat{r}}{r^2}$$ Where ##\vec{m}## is just a vector, and ##\hat{r}## is the unit radial vector and ##r## is the length of the radial vector. Homework Equations On the Levi Civita symbol...

Is this just a coincidence that cross product can be found from determinant of 3*3 matrix? what is the differences between wedge product and cross product?Thanks.

A fluid motion has velocity $$\underline{u}=\sin{(at)}\hat{\imath}+\hat{\jmath} \times r +\cos{(at)}\hat{k}$$ I need to know what is $$\hat{\jmath} \times r$$ to find Vorticity and other things.

Under what conditions is the common eigenspace of two commuting hermitian operators isomorphic to the direct product of their individual eigenspaces? As I'm not being able to precisely phrase my doubt, consider this example: Hilbert space of a two dimensional particle is the direct product of...

Some words before the question. For two smooth manifolds M and P It is true that T(M\times P)\simeq TM\times TP If I have local coordinates \lambda on M and q on P then (\lambda, q) are local coordinates on M\times P (right?). This means that in these local coordinates the tanget vectors are...

Homework Statement So a kaon moving at some speed in the +x direction spontaneously decays into one pion and one anti-pion. The anti-pion moves away with velocity of 0.8c, and the pion moves away with velocity of 0.9c. Mass of kaon = 498 MeV/c^2 Mass of pion/anti-pion = 140 MeV/c^2...

Dear friends, I have been told that if ##\{a_n\}_{n\in\mathbb{N}}##, ##\{a_{-n}\}_{n\in\mathbb{N}^+}##, ##\{b_n\}_{n\in\mathbb{N}}## and ##\{b_{-n}\}_{n\in\mathbb{N}^+}## are absolutely summable complex sequences -maybe even if only one i between ##\{a_n\}_{n\in\mathbb{Z}}## and...

Is there an intuitive reason or proof demonstrating that in general dimensions, there is no direct analogue of the binary cross product that yields specifically a vector? I came across Wedge Product as the only alternative, but am just learning linear algebra and don't quite comprehend yet...

HEY GUYS! (Wave) ok so i have this question i did. and now I am reviewing for the test and i looked at how i did it and i did in the most complicated way ever. i don't FULLY understand chegg's method. so i hope someone can provide me with the SIMPLEST method possible. thank u! (Blush) (p.s...

Hi, how can I compute the general solution of a system of linear equations? Non-square systems for example. I have the book Linear Algebra via exterior products, but it is the worst book in the history of math books, I think I'll burn it somehow, whatever. I can calculate the solution with the...

Hello! I have a problem in my calculus based physics class regarding vectors. The problem says: Vectors A and B have a scalar product -6.00 and their vector product has magnitude 9.00 what is the angle between these two vectors? Here is how I approached it: -6=|A||B|cos (theta) 9=|A||B|sin...

I'm asked to prove the following using Levi-Civita/index notation: (\mathbf{a \times b} )\mathbf{\times} (\mathbf{c}\times \mathbf{d}) = [\mathbf{a,\ b, \ d}] \mathbf c - [\mathbf{a,\ b, \ c}] \mathbf d \ I'm able to prove it using triple product identities, but I'm completely stuck...

So I've been working on physics homework and we have some vector/dot product questions. This is really long, but the questions I have really are rudimentary at best. I have seven total questions. You're given two vectors that only have an x and y component, A, and B, and the positive Z axis is...

I'm just trying to understand from a linear algebra standpoint how they define dot product from the inner product and how this gives rise to a definition of length and angle. somehow there is a way to combine points in space to a scalar value that unambiguously determines length and angle? Is...

Homework Statement Let ##G=G_{12}##, ##H_1=G_3##, ##H_2=G_2##. Decide if there are groups ##K_1##, ##K_2## such that ##G## can be expressed as the internal semidirect product of ##H_i## and ##K_i##.The Attempt at a Solution Suppose I can express ##G_{12}## as an internal semidirect product...

Homework Statement Hello, I've been trying to solve this problem, but in the examples that my teacher gave me didn't include something like this, I know how to calculate area but only if I have all the coordinates established. I need to find the area using the cross product. Homework...

Homework Statement http://postimg.org/image/lgphyvggz/ Homework Equations The Attempt at a Solution can someone explain where that transpose came from in (3.3)?

when differentiating e^(at) * (cos(bt) + isin(bt)) are you able to use product rule to find the derivative considering (cos(bt) + sin(bt)) as one function?? why?? and what does d/dt exactly mean?? (they get multiplied to a function that needs to be differentiated and I wanted to...

Hi, I just want to see if I understood this. Since the geometric product is associative and so on we can write for two multivectors A and B given by A= \alpha_{0}+\alpha_{1}e_{1}+\alpha_{2}e_{2}+\alpha_{3}e_{1}\wedge e_{2} B= \beta_{0}+ \beta_{1}e_{1}+\beta_{2}e_{2}+\beta_{3}e_{1}\wedge e_{2}...

1. Prove a) r=(u*v)=r*u+r*v and b) d/dt(r*s)=r*ds/st+dr/dt*s 2. Homework Equations : b) dr/dt=lim t->0=Δr/Δt and Δr=r(t+Δt)-r(t) 3. Attempt at the solution: Okay, so I was able to work out part a but I'm not quite sure how to start part b. Could anyone point me toward a useful resource to...

Homework Statement A and B are two unit vectors in the x-y plane. A = <cos(a), sin(a)> B = <cos(b), sin(b)> I need to derive the trig identity: sin(a-b) = sin(a) cos(b) - sin(b) cos (a) I'm told to do it using the properties of the cross product A x B Homework Equations A x B =...

Homework Statement Vectors A & B lie in an xy plane. A has a magnitude 7.4 and an angle 142(deg) with respect to the +x direction. B has components (-6.84i, -7.37j) B) What is the angle between the -y axis and the direction of the Cross product between A and B? Homework Equations Cross...

Hi there. I was following a deduction on continuum mechanics for the invariant nature of the first two laws of thermodynamics. The thing is that this deduction works with an identity, and there is something I'm missing to get it. I have the vector product: ##\vec \omega \times grad \theta##...

Let $0<a_i<\pi$, $i=1,\,\cdots,\,n$ and let $a=\dfrac{a_1+\cdots+a_n}{n}$. Prove that $\displaystyle \prod_{i=1}^{n} \left(\dfrac{\sin a_i}{a_i}\right)\le \left(\dfrac{\sin a}{a}\right)^n$.

Hey JO, I'm reading a book on geometric algebra and in the beginning (there was light, jk) a simple calculation is shown: Geometric product is defined as: ab = a \cdot b + a \wedge b or ba = a \cdot b - a\wedge b Now (a\wedge b)(a \wedge b)=(ab-a \cdot b)(a\cdot b - ba) =-ab^{2}a-(a...

Alright, so I was reading up on tensors and such with non-Cartesian coordinate systems all day but now I'm a bit tired an confused so you'll have to forgive me if it's a stupid question. So to express the dot product in some coordinate system, it's: g(\vec{A}\,,\vec{B})=A^aB^bg_{ab} And, if...

I am reading Chapter 2: Vector Spaces over \mathbb{Q}, \mathbb{R} \text{ and } \mathbb{C} of Anthony W. Knapp's book, Basic Algebra. I need some help with some issues regarding the general UMP-based definition of external and internal direct products ... ... On page 63, Knapp defines...

So, is there anyway to make the dot product change linearly? What I mean by this is when the angle is 45 degrees, I want it to be 0.5 instead of 0.7071 as you can see in this image: Instead I want 45 degrees to be 0.5, 60 degrees to be 0.33 and 30 degrees to be 0.66. Same would apply for...

Hello people. I'm thinking of using something like Twaron or Nomex fabric for a new application but one of the draw backs is the heat insulation property of these fabrics and I need to get around that and hopefully (in a wish upon a star kind of way) add a bit to the heat it can take. I was...

Hey guys, I was just doing some independent study on products of series and I'm trying to understand/derive the following form of the Cauchy product of series: \left(\sum_{n=0}^{N} a_{n}\right) \left(\sum_{m=0}^{N} b_{m}\right) = \sum_{n=0}^{N} \left(\sum_{k=0}^{n} a_{k}b_{n-k}\right)...

I'm reading through Douglas Gregory's Classical Mechanics, and at the start of chapter 6 he says that m \vec{v} \cdot \frac{d\vec{v}}{dt} = \frac{d}{dt}\left(\frac12 m \vec{v} \cdot \vec{v}\right), but I'm not sure how to get the right hand side from the left hand side. If someone could point...

Homework Statement Find x x = (1 + \frac{1+i}{2})(1 + (\frac{1+i}{2})^{2})(1 + (\frac{1+i}{2})^{2^{2}})...(1 + (\frac{1+i}{2})^{2^{n}}) Homework Equations Complex algebra equationsThe Attempt at a Solution Me again and another olympic question. I've tried some trigonometric substitutions...

I would like to convert: \frac{1}{1+x^{2m}} into a sum of terms. Preferably m terms but 2m terms would be OK. I start off with: \frac{1}{1+x^{2m}}=\frac{1}{\Pi^k_1(x-e^{i\frac{2k-1}{2m}\pi})(x-e^{-i\frac{2k-1}{2m}\pi})}=\frac{1}{\Pi^k_1 (x^2-2x \cdot cos \frac{2k-1}{2m}\pi +1)} where k = 1...

Hi all, I'm trying (and failing miserably) to understand tensors, and I have a quick question: is the inner product of a rank n tensor with another rank n tensor always a scalar? And also is the inner product of a rank n tensor with a rank n-1 tensor always a rank n-1 tensor that has been...

Hello, I'm working with a problem in linear elasticity, and I have to calculate the strain energy function as follows: 2W = σijεij Where σ and ε are symmetric rank 2 tensors. For cartesian coordinates it is really easy because the metric is just the identity matrix, hence: 2W = σxxεxx +...

Hi there. I wanted to demonstrate this identity which I found in a book of continuum mechanics: ##curl \left ( \vec u \times \vec v \right )=div \left ( \vec u \otimes \vec v - \vec v \otimes \vec u \right ) ## I've tried by writting both sides on components, but I don't get the same, I'm...

The uuu hadron doesn't violate Pauli's exclusion principle presumably because there is color. But even without color, can't the uuu exist if spatial wavefunctions are different? Suppose one u quark is located at r1, another at r2, and another at r3, and say that all three u quarks have spin up...

Homework Statement Suppose \vec{u}, \vec{v} and \vec{w} are vectors in an inner product space such that: inner product: \vec{u},\vec{v}= 2 inner product: \vec{v},\vec{w}= -6 inner product: \vec{u},\vec{w}= -3 norm(\vec{u}) = 1 norm(\vec{v}) = 2 norm(\vec{w}) = 7 Compute...

While studying Yang-Mills theory, I've come across the statement that there exists a positive-definite inner product on the lie algebra ##\mathfrak g## iff the group ##G## is compact and simple. Why is this true, and how it is proved?

With groups, one often seeks to create larger groups out of smaller groups, or the reverse: break down large groups into easier-to-understand pieces. One construction often employed in this regard is the direct product. The normal way this is done is like so: The direct product of two groups...

Hey guys I'm new here and I've been using MathType for all on-screen math. For some reason PF doesn't recognize the built-in Dirac's bra-ket notation. (i.e. <\psi|) So I've included my equations and solution in the format of images, hopefully it isn't a problem. Homework Statement Proof...

Definition/Summary The product rule is a method for finding the derivative of a product of functions. Equations (fg)'\ =\ f'g\ +\ fg' (fgh)'\ =\ f'gh\ +\ fg'h\ +\ fgh' Extended explanation If a function F is the product of two other functions f and g (i.e. F(x) = f(x)g(x))...