Product Definition and 1000 Threads

Can I say that \eta^{ij}\eta_{km}=\delta^{i}_{k}\delta^{j}_{m}? Kind of in the same way that they yield one delta in the case where one of their indices is summed over? Thanks

Hello I have a set of sets of real numbers greater than 1. Each set can have a different quantity of numbers. Set A1 {a11, a12,...a1m1} Set A2 {a21, a22, ..., a2m2} ... Set AN {aN1, aN2, ..., aNmN} If I want the sum of all possible products that have one element from each set, that's...

The Vector A points 17° counterclockwise from the positive x axis. Vector B lues in the first cuadrant of the xy plane. The magnitudes of the cross product and the dot product are the same: i.e, |AXB|= |A(times)B| What Angle does B make with the positive x axis? 2. Is ti a scalar...

Consider the function APL=\frac{\sqrt[4]{L}}{L}, where L is the number of workers. The company has just hired 8 workers. What is the marginal product of the labor?I know that if I had the total product I could differentiate it and get the marginal product, but it's the average product that is...

Dear Forum : I hung up with a integration http://ppt.cc/mIpV Can it be deduced to a simpler form? The distribution of σ(E) is http://ppt.cc/-5Z5 The estimation width of x is 10MeV , height is 200mb. The distribution of dE/dx is http://ppt.cc/vcVU Is there a way to do some simple...

Homework Statement Assume A and B are normal linear operators [A,A^{t}]=0 (where A^t is the adjoint) show that det AB = detAdetB Homework Equations The Attempt at a Solution Well I know that since the operators commute with their adjoint the eigenbases form orthonormal sets...

I was working on a pde, and I needed to compute a Jacobian for it. Suppose we have a function consisting of a series of matrices multiplied by a vector: f(X) = A * B * b --where X is a vector containing elements that are contained within A, b, and/or b, --A is a matrix, B is a matrix, and b is...

Hi, So I still not sure how to apply like rhr rule in this setup in problem like the one in the following so I tried to do rhr in order to get the direction but it didn't work out. this is an example from halliday and resnick book. Figure 32-24 shows a wire segment,placed in a uniform...

Just for fun, eh...? (Heidy)For $$z \in \mathbb{R}$$, and $$m \in 2\mathbb{N}+1$$, show that:$$\frac{\tan mz}{\tan z}=\prod_{j=1}^{ \lfloor m/2 \rfloor } \tan\left(\frac{j\pi}{m}+z\right) \tan\left(\frac{j\pi}{m}-z\right) $$

I was wondering about the proper way to say, \langleA|B\rangle . I have recently heard, "The inner product of A with B." But I'm not sure if this is correct. Does anyone know the proper order in which to place A and B in the sentence? As a simple example: Suppose you're speaking with...

A simple question: what is the difference between inner product and dot product?

Suppose I was asked if G \cong H \times G/H . At first I considered a familiar group, G = S_3 with its subgroup H = A_3 . I know that the quotient group is the cosets of H, but then I realized that I have no idea how to interpret a Cartesian product of any type of set with elements that aren't...

Let $$f:\mathbb{R} \to \mathbb{R}$$ and $$g:\mathbb{R} \to \mathbb{R}$$ be discontinuous at a point $$c$$ . Give an example of a function $$h(x)=f(x)g(x)$$ such that $$h$$ is continuous at c. $$ f(x) = \begin{cases} 0 & \text{if } x \in \mathbb{Q} \\ 1 & \text{if } x \in...

Homework Statement For what values of k is (scalar product of vectors a and b) = a_{1}b_{1}-a_{1}b_{2}-a_{2}b_{1}+ka_{2}b_{2} a valid scalar product? The vectors a and b are defined as: a = a_{1}e_{1} + a_{2}e_{2} b = b_{1}e_{1} + b_{2}e_{2} where e_{1} and e_{2} are unit vectors...

I am currently going through the book Introduction Of Electrodynamics by Griffiths. I have come across vector triple product which is stated as follows in the book: $$\textbf{A} \times (\textbf{B} \times \textbf{C})=\textbf{B}(\textbf{A}\cdot \textbf{C})-\textbf{C}(\textbf{A}\cdot...

Let's say we have operator X that is Hermitian and we have operator P that is Hermitian. Is the following true: [X,P]=ihbar This is the commutator of X and P. This particular result is known as the canonical commutation relation. Expanding: [X,P]=XP-PX=ihbar This result indicates that...

Hi, I was looking for a proof or explanation of this. From Schroeder's Thermal Physics, pg 56, explaining interacting systems in equilibrium. The example in the text is two 3-harmonic oscillators with a total of 6 units of energy. So one macrostate is where each has 3 units of energy. The...

We work on optical simulation where we use not ideal beam expander. Not ideal means that for beam expander designed for single mode (M^2=1), the output beam has M^2 >1 (M^2 = M squared) In our system we want to use beam expander with multimode laser beam. The beam expander is not ideal (for...

Hi, can somebody help me with the problem: Suppose that in a vector space over field of real numbers a positive defined norm is defined for each vector which satisfies the triangle inequality and ||aU||=|a|*||u||. Show that a real valued scalar product can de defined as follows...

Hello, I have a quick question about integrals of dot products. We are learning about magnetic flux as the integral of b dot da. However, what circumstances must be present where we can simplify this integral into (b*a) and ignore the integral?

What is the difference between a dot product and an inner product. The internet says that they are generalizations of each other. What does that even mean? Thanks for any help.

Homework Statement Let \left[a,b\right] be a closed bounded interval, f : [a,b] \rightarrow \textbf{R} be bounded, and let g : [a,b] \rightarrow \textbf{R} be continuous with g\left(a\right)=g\left(b\right)=0. Let f_{n} be a uniformly bounded sequence of functions on \left[a,b\right]. Prove...

How does one change the dot product such that there is no dot product in between, just plain multiplication? For example, in the following: eb.\partialcea=-\Gammaa bc How do I get just an expression for \partialcea?

I would like to know why $M_n$ $\not\cong$ $O_n$ x $T_n$, where $M_n$ is the group of isometries of $\mathbb R^n$, $O_n$ is the group of orthogonal matrices, and $T_n$ is the group of translations in $\mathbb R^n$. **My attempt:** Can I show that one side is abelian, while the other group is...

Homework Statement Prove that if u and v are nonzero vectors, and theta is the angle between them then u dot product v = ||u|| ||v|| cos (theta). Consider the triangle with sides u ,v , and u-v. The Law of Cosines implies that ||u-v||^2 = ||u||^2 + ||v||^2 - 2||u|| ||v|| cos(theta). On the...

I got asked how the scalar product can be used to find the length of a vector? Could someone please explain

Find the product of 3 distinct real numbers $a, b, c$ if they satisfy the system of equations $a^3=3b^2+3c^2-25$ $b^3=3c^2+3a^2-25$ $c^3=3a^2+3b^2-25$

Homework Statement See attached image Homework Equations The Attempt at a Solution For the first half of the question, ordered pairs would be (1, [1]), since 1 and [1] are the multiplicative identities in these rings. but no matter how many times we add (1, [1]) to itself...

Correct me if I'm wrong here but it is my understanding that vector spaces are given structure such as inner products, because it allows us to use these structured vector spaces to describe and analyse physical things with them. So physical properties such as 'distance' cannot be analysed in...

Let A be a Hermitian operator with n eigenkets: A|u_i\rangle = a_i |u_i\rangle for i=1,2,...,n. Suppose B is an operator that commutes with A. How could I show that \langle u_i | B | u_j \rangle = 0 \qquad (a_i \neq a_j)? I have tried the following but not sure how to proceed: AB -...

Let $p$ be the sum and $q$ be the product of all real roots of the equation $x^4-x^3-1=0$. Prove that $q<-\dfrac{11}{10}$ and $p>\dfrac{6}{11}$.

Warning This product attracts every other piece of matter in the Universe, including products of other manufacturers, with a force proporitional to the product of the masses and inversely proportional to the square of the distance between them.Handle with Extreme Care This product contains...

Homework Statement I need to answer a bunch of topological questions based on the cartesian product of two sets, but I'm not entirely sure how to graph them out. I have A = [1,2)U{3} and B = {1, (1/2), (1/3), ...}U[-2,-1). S = A x B, and I need the graph of S. Could anyone help me with...

Consider the following commutator for the product of the creation/annihilation operators; [A*,A] = (2m(h/2∏)ω)^1 [mωx - ip, mωx + ip] = (2m(h/2∏)ω)^1 {m^2ω^2 [x,x] + imω ([x,p] - [p,x]) + [p,p]} Since we have the identity; [x,p] = -[p,x] can one assume that.. [x,p] - [p,x] =...

Need someone to check my answer please. Consider a 4 input, 1 output digital system (W,X,Y,Z, and f respectively) . Design a POS circuit with any number of inputs such that f(W,X,Y,Z) = M(0,2,4,9,13) + D(6,14). First fill in the Truth Table, then find the minimum product of sums equation using...

can you show me derive cross product from dot product?

hey all well the title says it all. if i want to take the cross product of two matrices, how do i do it? any help, advice, etc. is very appreciated! thanks

How do you take the cross product of two 4-Vectors? \vec{r} = \left( \begin{array}{ccc}c*t & x & y & z \end{array} \right) \vec{v} = \left( \begin{array}{ccc}c & vx & vy & vz \end{array} \right) \vec{v} \times \vec{r} = ?

Homework Statement Use (the bac-cab rule) to expand this triple product:L = mr x (ω x r) If r is perpendicular to ω, show that you obtain the elementary formula, angular momentum = mvr. (The bold letters are vectors.) Homework Equations A X (B X C) = (A\cdotC)B - (A\cdotB)C...

Homework Statement Compound A mass - 11.0g solubility in ethanol at 78 deg C - 0.53 g/mL solubility in ethanol at 0 deg C - 0.08 g/mL Impurity B mass - 2.5 g solubility in ethanol at 78 deg C - 0.82 g/mL solubility in ethanol at 0 deg C - 0.12 g/mL Impurity C mass - 2.5 g solubility in...

Show that $3^{2008}+4^{2009}$ can be written as product of two positive integers each of which is larger than $2009^{182}$

Homework Statement Hi all, Here's the problem: Prove, in tensor notation, that the triple scalar product of (A x B), (B x C), and (C x A), is equal to the square of the triple scalar product of A, B, and C. Homework Equations The Attempt at a Solution I started by looking at the triple...

Show that TS^1 is diffeomorphic to TM×TN. (TS^1 is the tangent bundle of the 1-sphere.) We can use the theorem stating the following. If M is a smooth n-manifold with or without boundary, and M can be covered by a single smooth chart, then TM is diffeomorphic to M×ℝ^n. Clearly, I must be...

Write out the minimal Product of Sums (POS) equation with the following Karnaugh Map. Just need someone to check my work please. I am questioning my self on my grouping. Did I group correctly or should I have grouped the bottom left 0 and D versus the 0 in the group of 8? Thanks for your time...

okay so I know that the area of the triangle is half the area of the parallelogram, ill try using pictures because this is a bit confusing to describe only with words: for example we have this http://farside.ph.utexas.edu/teaching/301/lectures/img243.png and then if we use the cross product of...

Awesome thanks.. Mind checking this as well? Minimize Sum of Products equation given the following K map. My Answer: $$\bar{y} \bar{w} + wx + y\bar{z}w + yw\bar{x} $$

Just need someone to check my work. Couldn't find the problem via Google. $f$(W,X,Y,Z) M (0,1,2,7,12,15) + d(3,13). 1)Find the minimum Product of Sums equation using a K-Map. 2)Draw a schematic of a minimized circuit implementing the logic using NOR gates. 1) My Answer: $$(\bar{w} +...

I'm following the book ''introduction to electrodynamics by D.J. Griffiths''. As he has written that the formula ''integral of product of infintesimal volume with the square of electric field'' gives us the energy contained in a charge configuration that is always positive because we're...

i have a vector xK where k is the unit vector perpendicular to other unit vectors i and j when i multiply a force which has 5k for instance another which has ( 3 i + 4 j ) i multiply 5k by 3i then 5k by 4j right ? the answer would be ( 15 j - 20 i ) right ?

Sorry guys, I have some differential topology homework, and I may be asking a lot of questions in the next few days. Problem Statement Suppose M_1,...,M_k are smooth manifolds and N is a smooth manifold with boundary. Then M_1×..×M_k×N is a smooth manifold with a boundary. Attempt Since...