Product Definition and 1000 Threads

Homework Statement http://photouploads.com/images/8ba21e.png {moderator's not: Inserted image so it's visible without having to follow a link} 2. Homework Equations a·(b x c) = b·(c x a) The Attempt at a Solution How did they get from the (b) to (c)? In particular, I am referring to the...

hi! i have some strange by-product, intermediate, or something... i prepared inorganic complex, and it is contaminated ith something (have problems with purification), when i measure platinum NMR, i can observe signal of my product (which should be yellow), but it is yellowish and white (or...

Hey guys, I got the following derivation for some physical stuff (the derivation itself is just math) http://thesis.library.caltech.edu/5215/12/12appendixD.pdf I understand everything until D.8. After D.7 they get the eigenvalue and eigenvectors from ε. The text says that my δx(t) gets aligned...

Homework Statement How would I find the X coordinate of point A. So far I have: A: ( X=? Y=-231.125" Z=175") B: (X=0" Y=0" Z=144") https://physicsforums-bernhardtmediall.netdna-ssl.com/data/attachments/90/90134-44f7db5f8e2c5989352d374160952d17.jpg Homework Equations...

Homework Statement Minimize the functional: ∫01 dx y'2⋅ ∫01 dx(y(x)+1) with y(0)=0, y(1)=aHomework Equations (1) δI=∫ dx [∂f/∂y δy +∂f/∂y' δy'] (2) δy'=d/dx(δy) (3) ∫ dx ∂f/∂y' δy' = δy ∂f/∂y' |01 - ∫ dx d/dx(∂f/∂y') δy where the first term goes to zero since there is no variation at the...

I have just begun reading about Einstein's summation convention and it got me thinking.. Is it possible to represent ∑aibici with index notation? Since we are only restricted to use an index twice at most I don't think it's possible to construct it using the standard tensors (Levi Cevita and...

Homework Statement This is not a homework problem, I am currently reading the Derivation of potential of a charged particle in Electric and Magnetic field from the book Mechanics by Symon (I attached the image of the page), I need to know how to expand the vector cross product such as...

Homework Statement Proof that if the slopes of two lines a1, a2 (that are not vertical), m1,m2 satisfy: m1*m2 = -1, then the lines are perpendicular. Homework EquationsThe Attempt at a Solution I tried to use the tan function, so that m1 = tanΘ where Θ1 is the angle of the line formed from x...

The 2-D plane is usually constructed as "ℝxℝ" and ℝ is both open and closed. My question is, what is the direct product of a half open and an open interval? Is it also open or half open?

Suppose we are given this definition of the wedge product for two one-forms in the component notation: $$(A \wedge B)_{\mu\nu}=2A_{[\mu}B_{\nu]}=A_{\mu}B_{\nu}-A_{\nu}B_{\mu}$$ Now how can we show the switch from tensor products to wedge product below...

Homework Statement Let A and B be n by n matrices such that A is invertible and B is not invertible. Then, AB is not invertible. Homework EquationsThe Attempt at a Solution We know that A is invertible, so there exists a matrix C such that CA = I. Then we can right -multiply by B so that CAB...

Problem: Fix some vector ##\vec{a} \in R^n \setminus \vec{0}## and define ##f( \vec{x} ) = \vec{a} \cdot \vec{x}##. Give an expression for the maximum of ##f(\vec{x})## subject to ##||\vec{x}||_2 = 1##. My work: Seems like a lagrange multiplier problem. I have ##\mathcal{L}(\vec{x},\lambda)...

Homework Statement I am asked to offer an example of two commuting elements whose product does not have an order equal to the least common multiple of their individual orders. Homework EquationsThe Attempt at a Solution Consider ##-1## and ##1## in ##\mathbb{Z}##. Then ##1+(-1) = 0## which...

Hello - I'm not sure this is where this should go, but I'm working with Laplace Transforms and differential equations, so this seems as good a place as any. Also, I doubt this is graduate level math strictly speaking, but I went about as high as you can go in calculus and linear algebra during...

The first part of the problem was easy enough as was finding the directions of the other two, but I am having trouble finding the correct angle for the last two cross products in order to find the magnitude.

I have a question about calculating solubilities of sparingly soluble salts. Eg Ksp CaF2 = 4 x 10-11 So, Saturation Index of CaF2 is: SICaF2 = IP / Ksp Where IP = Ionic Product = [Ca2+] x [F-]2 [Ca2+] and [F-] are molar concentrations of each ion. Example: We have 400 ppm Ca and 12 ppm...

Homework Statement Let ##A## be an n × p matrix and ##B## be an p × m matrix with the following column vector representation, B = \begin{bmatrix} b_1 , & b_2, & ... & ,b_m \end{bmatrix} Prove that AB = \begin{bmatrix} Ab_1 , & Ab_2, & ... & , Ab_m \end{bmatrix} If ##A## is represented...

Homework Statement A\cdot B\times C\quad =\quad 2\\ (2A+B)\quad \cdot \quad [(A-C)\quad \times \quad (2B+C)]\quad =\quad ? Homework Equations Various cross product and dot product properties The Attempt at a Solution I've only managed to get so far, don't really know what to do next A\cdot...

Hi, For school we are currently working with heterogeneous equilibria. I am given a salt that will be solved in water and I have to calculate the concentrations of the ions. I have to use the solubility product for this. In the Netherlands we are provided with a reference book that has all...

I am trying to derive the law of signs from the cross product. First, we have three vectors ##\vec{A} ~\vec{B} ~\vec{C}## such that ##\vec{A} + \vec{B} + \vec{C} = 0##. This creates a triangle. Then, we label the angles opposite the respective sides as a, b, and c. I am not sure where to go...

Homework Statement Verify the identity: ## \nabla \times ( A \times B) = (B\bullet \nabla)A - (A\bullet\nabla)B + A(\nabla \bullet B)-B(\nabla\bullet A)## My issue here is I don't understand the significance of why a term has B or A on the left of the dot product, and another has B or A on...

I recently learned that the general formula for the dot product between two vectors A and B is: gμνAμBν Well, I now have a few questions: 1. We know how in Cartesian coordinates, the dot product between a vector and itself (in other words A ⋅ A) is equal to the square of the magnitude |A|2...

Homework Statement Hello, We know that if A and B are two unbounded densely defined operators, it does not mean that AB is also densely defined. But if A is bounded then D (AB) = D (B) ie AB is densely defined. Is AB densely defined if: 1) B is bounded and A is unbounded densely defined...

Scalar Product is defined as ##\mathbf A \cdot \mathbf B = | \vec A | | \vec B | \cos \theta##. With the construct of a triangle, the Law of Cosines is proved. ##\mathbf A## points to the tail of ##\mathbf B##. Well, ##\mathbf C## starts from the tail of ##\mathbf A## and points to somewhere...

I have seen a proof for the formula of A.B = ||A|| ||B|| cos(theta)[ proof using the diagram and cosine rule]. In the proof they have assumed that distributive property of dot product is right. diagram is given below c.c =(a-b).(a-b) = a^2 +b^2 -2(a.b) [ here they used distributive law] I...

Dear all, I know how to interpret a vector, inner product etcetera in one Hilbert space. However, I can not get my head around how the direct product of two (or more) Hilbert spaces can be interpreted. For instance, the Hilbert space ##W## of a larger system is spanned by the direct product of...

Hi. I'm trying to understand tensors and I've come across this problem: "Show that, in general, a (2, 0) tensor can't be written as a tensor product of two vectors". Well, prior to that sentence, I would have thought it could... Why not?

Why A .A =||A||^2(dot product) in vector analysis. Every where in vector analysis mathematicians used this result. To prove A.B = ||A|| ||B||cos(theta) we assume that A.A is ||A||^2 , without assuming this we can't prove A.B = ||A|| ||B||cos(theta) . I think they assumed it because dot product...

Homework Statement Consider the vector space of all continuous functions on the interval C[-1,1]. That is V = C[-1,1] show that <f(x),g(x)> = ∫(-1,1) x2f(x)g(x)dx defines an inner product on C[-1,1]. I have shown <f,g> = <g,f>, <kf,g> = k<f,g>, <f+g,h> = <f,h> + <g,h> and I am trying to...

Why mathematicians defined that the cross product of vector A and B will be a vector perpendicular to them.

I do not know if this is the proper rubric to ask this question, but I picked the one that seemed the most relevant. I have noticed some superficial resemblance between the tensor product and the ultraproduct definitions. Does this resemblance go any further? While I am on the subject of...

Prove \left(1+\frac{1}{\sin x}\right)\left(1+\frac{1}{\cos x}\right)\gt 5 for 0\lt x \lt \frac{\pi}{2}.

Are there such companies that specialize in making existing products reliable and safe. Thanks. P.S. I have a patent that I need to improve.

svein submitted a new PF Insights post Reflections on Product Quality Continue reading the Original PF Insights Post.

The problem: By considering w x (p x w) resolve vector p into a component parallel to a given vector w and a component perpendicular to a given vector w. Hint: a x (b x c) = b(a x c) - c(a x b) I'm afraid I really have no idea where to go with this one. The hint leads to: p(w.w) - w(w.p) =...

Homework Statement Under which set of conditions is the ionic product of water, Kw, constant at a given temperature in aqueous systems? in dilute acidic but not dilute alkaline solutions.in dilute alkaline but not dilute acidic solutions.in both dilute acidic and alkaline solutions.only at...

If I have 2 complex numbers, A and B, what is the correct way to evaluate this expression: ## E = AB - B^*A^*## I was under the impression that when taking the product of complex numbers, you always conjugate one factor, but in this instance, it is quite important which one is conjugated, no...

Hi. Why did the founding fathers of QM know that the Hilbert space of a composite system is the tensor product of the component Hilbert spaces and not a direct product, where no entanglement would emerge? I mean today we can verify entanglement experimentally, but this became technologically...

Homework Statement This is a child thread I'm creating from a previous topic: https://www.physicsforums.com/threads/combinatorics-problem.871661/#post-5473920 In that thread, I was helped to come up with the expression for the number of arrangements of R distinct types of objects given the...

We had integrals, so we have to have series as well. Here are 10 easy to difficult series and infinite products. Up to you to find out the exact sum. Rules: The answer must be a finite expression. The only expressions allowed are integers written in base 10, the elementary arithmetic...

Hello! I am reading something about applications of group theory in quantum mechanics and I got confused about the difference between direct sum and direct product. In many places I found that they mean the same thing. However, the ways I found them defined in the book I read from, seem to be...

Homework Statement δ(z*-z0*)δ(z+z0)=? δ(z*+z0*)δ(z-z0)=? where 'z' is a complex variable 'z0' is a complex number Formula is just enough, derivation is not needed.

Homework Statement What is the product of two Dirac delta functions δ(Real(z-c))δ(Img(z-c))=? 'z' and 'c' are complex numbers. This is not a problem, But I just need to use this formula in a derivation that I am currently doing. I just want the product of these two Dirac delta functions as a...

Homework Statement This problem actually has 2 parts. For vectors a= (2,p,8) and b= (q,4,12), determine the values of p and q so that the vectors are a)perpendicular b) collinear textbook answer: a) p= 1 and q= -50 (answers may vary) b) p= 8/3 and q=3 Homework Equations a*b = 0 a*b =...

if for rational a,b,c $ax^3+bx+c=0$ one root is product of 2 roots then that root is rational

Homework Statement Using index-comma notation only, show: \begin{equation*} \underline{\bf{v}} \times \text{curl } \underline{\bf{v}}= \frac{1}{2} \text{ grad}(\underline{\bf{v}} \cdot \underline{\bf{v}}) - (\text{grad } \underline{\bf{v}}) \underline{\bf{v}} \end{equation*} Homework Equations...

Homework Statement Using index notation only (no expanding of terms), show: \begin{equation*} \text{(a) }\underline{ \bf{a}} \cdot \underline{\bf{A}} \underline{\bf{b}} = \underline{\bf{A}} \cdot \underline{\bf{a}} \otimes \underline{\bf{b}} \end{equation*} Homework Equations...

Homework Statement Using index notation only (i.e. don't expand any sums) show that: \begin{align*} &\text{(a) } \epsilon_{ijk} \det \underline{\bf{A}} = \epsilon_{mnp} A_{mi} A_{nj} A_{pk} \\ & \text{(b) } \det \underline{\bf{A}} = \epsilon_{mnp} A_{m1} A_{n2} A_{p3} \end{align*} Homework...

I was reading about Gauss's Lemma here: https://cims.nyu.edu/~kiryl/Algebra/Section_3.10--Polynomials_Over_The_Rational_Field.pdf Unfortunately, I am stuck on Lemma 3.10.1 that concludes that the product of a pair of primitive polynomials is itself primitive. I understand about how there is...

I am reading Donald S. Passmore's book "A Course in Ring Theory" ... I am currently focussed on Chapter 9 Tensor Products ... ... I need help in order to get a full understanding of the free abelian group involved in the construction of the tensor product ... ... The text by Passmore...