Product Definition and 1000 Threads

For part a, We have ##α+β=b## and ##αβ =c##. It follows that, ##(α^2 + 1)(β^2+1)=α^2β^2+α^2+β^2+1)## =##α^2β^2+(α+β)^2-2αβ +1## =##c^2+b^2-2c+1## =##c^2-2c+1+b^2##...

My apologies if this question is trivial. I have searched the forum and haven't found an existing answer to this question. I've been working through differential geometry problem sets I found online (associated with MATH 481 at UIUC) and am struggling to show that T(MxN) is diffeomorphic to TM...

When I look at a range of inputs around x=c and consider the corresponding range of outputs If 0< |x-c| <δ -> |f(x)-L1|<ϵ1 and |g(x)-L2|<ϵ2 as we shrink the range of inputs the corresponding outputs f(x) and g(x) narrow on L1 and L2 respectively. |f(x)-L1||g(x)-L2|<ϵ2ϵ1 The product of the...

Freezable. Maintains flexibility when frozen. Re-usable. Non toxic. Need to know the chemical name to enable sourcing for a manufacturer.

In Gravitation by Misner, Thorne and Wheeler (p.139), stress-energy tensor for a single type of particles with uniform mass m and uniform momentum p (and E = p2 +m2) ½ ) can be written as a product of two 4-vectors,T(E,p) = (E,p)×(E,p)/[V(E2 – p2 )½ ] Since Einstein equation is G = 8πGT, I am...

I am sure you are all familiar with the cross product in 3D space. i cross into j gives k. Cyclic Negative, if reversed, etc. I am sure you are all familiar with the definition as: norm of the first vector, norm of the second, sine of the angle, perpendicular (but direction using right hand...

Using the inequality of arithmetic and geometric means, $$\frac {x+y}{2}≥\sqrt{xy}$$ $$6^2≥xy$$ $$36≥xy$$ I can see the textbook answer is ##36##, my question is can ##x=y?##, like in this case.

##f## is continuou on ##\mathbb{C}##, so for al ##\epsilon>0##, there is a ##\delta>0## such that $$|\tilde{z}-z|\leq \delta \Rightarrow |f(\tilde{z})-f(z)|\leq \epsilon$$ for all ##\tilde{z}## and ##z## in ##\mathbb{C}##. Complex conjugation is a norm preserving operation on ##\mathbb{C}##, so...

I'm learning Linear Algebra by self and I began with Apsotol's Calculus Vol 2. Things were going fine but in exercise 1.13 there appeared too many questions requiring a strong knowledge of Real Analysis. Here is one of it (question no. 14) Let ##V## be the set of all real functions ##f##...

Why F=m*a i.e product or multiplication and not F=m+a? i.e addition or summation?

Dear PF, so we know that cross product of two vectors can be permutated like this: ## \vec{ \alpha } \times \vec{ \beta }=-\vec{ \alpha} \times \vec{ \beta} ## But in a specific case, like ## \vec{p} \times \vec{A} = \frac{ \hbar }{ i } \vec{ \nabla } \times \vec{A} ## the cyclic permutation of...

Although it is considered unwise to judge a book by its cover, a book's cover is still useful for finding the direction of the cross product ##\mathbf{A}\times \mathbf{B}## between two given vectors. Being able to read is all that is needed. Here is a detailed procedure. Step 1. Move one...

I tried to find the components of the vectors. ##a_y =2.60 sin 63.0 = 2.32## and assuming the z axis would behave the same as an x-axis ##a_z =2.60 cos 63.0 = 1.18## ##b_z =1.30 sin 51.0 = 1.01## making the same assumption ##b_x =1.3 cos 51.0 = 0.82## I now think I should have switched these...

We denote a scalar product of two vectors ##a, b## in Hilbert space ##H## as $(a,b)$. In Bra Ket notation, we denote a vector a in Hilbert space as ##|a\rangle##. Also we say that bras belong to the dual space ##H##∗ . So Bras are linear transformations that map kets to a number. Then it...

Summary:: I need to solve a problem for an assignment but just couldn't find the right approach. I fail to eliminate b or c to get only the magnitude of a. Let a, b and c be unit vectors such that a⋅b=1/4, b⋅c=1/7 and a⋅c=1/8. Evaluate (write in the exact form): - ||4a|| - 3a.5b - a.(b-c) -...

Hi If i have 2 general vectors written in Cartesian coordinates then the scalar product a.b can be written as aibi because the basis vectors are an orthonormal basis. In Hamiltonian mechanics i have seen the Hamiltonian written as H = pivi - L where L is the lagrangian and v is the time...

My notes says that the geometrical meaning of $$|\vec v \times \vec w | $$ is the perpendicular distance from point ##V## to line passing through ##O## and ##W## (all vectors are position vectors) $$|\vec v \times \vec w | = |\vec v| |\vec w| \sin \theta$$ From the picture, the perpendicular...

I am trying to find the equations of motion of the angular momentum ##\boldsymbol L## for a system consisting of a particle of mass ##m## and magnetic moment ##\boldsymbol{\mu} \equiv \gamma \boldsymbol{L}## in a magnetic field ##\boldsymbol B##. The Hamiltonian of the system is therefore...

> **Exercise.** Let T1and T2be tensors of type (r1 s1)and (r2 s2) respectively on a vector space V. Show that T1⊗ T2can be viewed as an (r1+r2 s1+s2)tensor, so that the > tensor product of two tensors is again a tensor, justifying the > nomenclature... What I’m reading：《An introduction to...

You are given the dollar value of a product in 2016 and the rate at which the value of the product is expected to change during the next 5 years. Use this information to write a linear equation that gives the dollar value V of the product in terms of the year t. (Let t = 16 represent 2016.)1...

Hi If i calculate the vector product of a and b in cartesian coordinates i write it as a determinant with i , j , k in the top row. The 2nd row is the 3 components of a and the 3rd row is the components of b. Does this work for sphericals or cylindricals eg . can i put er , eθ , eφ in the top...

The unormalised plane wave solution is given as ##u_{\vec{k}}=e^{i\vec{k}\cdot\vec{x}-i\omega t}##. I want to show that ##(u_{\vec{k}},u^{*}_{\vec{k}'})=0##. However, I don't seem to be able to get the answer through direct calculation. Any hints on how to obtain the answer?

I am confused. Why sometimes perturbation ##V'=\alpha xy## we can write as ##V'=\alpha x \otimes y##. I am confused because ##\otimes## is a tensor product and ##x## and ##y## are not matrices in coordinate representation. Can someone explain this?

Hey! :giggle: For $p\in \mathbb{R}^2$ let $\delta_{p,\alpha}=\tau_p\circ \delta_{\alpha}\circ\tau_p^{-1}$. Let $p,q\in\mathbb{R}^2$ and $\alpha,\beta\in \mathbb{R}$. (a) Show that $\gamma=\delta_{p,\alpha}\circ\delta_{q,\beta}$ is a rotation of a translation (or both). Give the center of...

v=197 If $y=x \sin x,$ then $\dfrac{dy}{dx}=$ $a.\quad\sin{x}+\cos{x}$ $b.\quad\sin{x}+x\cos{x}$ $c.\quad\sin{x}+\cos{x}$ $d.\quad x(\sin{x}+\cos{x})$ $e.\quad x(\sin{x}-\cos{x})$ well just by looking at it because $dx(x) = 1$ elimanates all the options besides b $1\cdot \sin (x)+\cos (x)x$...

I have to do inverse laplace transform of infinite product that is shown below. Can somebody help me with that?

Hi guys, I am losing my mind over this passage... I cannot understand how to get from the first expression with the cross products to the second ##\dot{\textbf{r}}(\textbf{r}\cdot \textbf{r})-\textbf{r}(\textbf{r}\cdot\dot{\textbf{r}})##

What I mean is the way that a product of cosines in which the angles increment the same amount is equal, with some extra terms, of the sum of the cosines. It is discussed here...

Let us consider the infinite products ## p_{n}\,=\, 2\cdot 2\cdot 2 \cdot 2 \cdots 2 \,=\, 2^n## with ##n=1,\ldots ## . Clearly ##p_{n}\rightarrow +\infty## as ##n\rightarrow +\infty##. But if I put the infinity case ## 2\cdot 2\cdot 2 \cdot 2 \cdots \,=\, x## I have ##2\cdot x =x ## so...

I have a lot of acorns in my pebbles. I'm looking for a way to sort these out quickly. I was thinking of the possibility that something floats on water and another material does not. If I use plain water, most of these acorns will sink too. So my questions is how much salt or other product...

I have included here the screen shot of the page I am referring to.I am unsure of how this non-local Lagrangian of Eqtn(32.68) has been constructed. Have they just integrated the interaction Lagrangian densities over two different sets of points (x & y) ? If so, then why is there no P_L in...

My book gives this formula for the semidirect product for groups ##Z_p## and ## Z_q## for primes p<q and p divides (q-1). ##(a,b)*(x,y)=(a+_q c^bx,b+_py)## There is also an explanation of what c is but very little else. It doesn't even explain what operation adjacency represents, eq...

Find all positive integers $k$ such that the product of the digits of $k$, in the decimal system, equals $\dfrac{25k}{8}-211$.

After finding the number of elements for this group, how do I extend the argument to $$p,q\equiv1\left(mod\ 3\right)$$, where $$G=(C_p:C_3\ )\times(C_q:C_3\ )$$Any help appreciated.

Hi, PF, I think I've found a typo in my textbook. It says: "In the case of a multiplication by a constant, we've got $$(Cf)'(x)=\displaystyle\lim_{h \to{0}}{\dfrac{Cf(x+h)-Cf(x)}{h}}=\displaystyle\lim_{h \to{0}}{\dfrac{f(x+h)-f(x)}{h}}=Cf'(x)$$" My opinion: it should be...

Prove that in the following product $P=(1-x+x^2-x^3+\cdots-x^{99}+x^{100})(1+x+x^2+x^3+\cdots+x^{99}+x^{100})$ after multiplying and collecting like terms, there does not appear a term in $x$ of odd degree.

The book I'm following (Gallian) basically says: r can't be 1 since then it won't map all elements to themselves. If r=2, then it's already even, nothing else to do. If r>2, Then consider the last two factors: ##\beta_{r-1} \beta_r##. Let the last one be (ab). Since the order of elements...

I'm trying to learn Group Theory from Gallian's book. When I reached the chapter for permutation groups, the author gives an example that we can write (12345) as (15)(14)(13)(12). I immediately recognized that this should always work (I proved it later.) Then author says we can write : (12345)...

I study from Gourgoulhon's text 'special relativity in general frames', I have some difficulty to understanding Chapter 3 Page 84. I already learn that there exist a orthogonal projection mapping ##\bot_{u}:E \rightarrow E_u(P)## from the vector space ##E \cong R^4## to the subspace ##E_u(P)##...

So I that I need to prove the axioms: associativity, existence of the identity element, and existence of the right inverse. For associativity I know that the binary operations of G and H have to already be associative, and the elements of G X H are made up of these binary operations, so...

Find the minimum value for $x$ where $x\in \mathbb{N} $ and $x^2-x+11$ can be written as a product of 4 primes, which are not necessarily distinct.

At constant pressure Enthalpy change is equal to heat exchange and we say that "if Enthalpy Change is negative then product formed is stable", I am not able to make sense of this statement as change in Enthalpy tells us only about heat exchange but internal energy is function of both Work and...

I am trying to simulate fission product ejection from thin fissile films in gas filled detectors (fission chamber). Does MCNP 6.2 produce recoil fission products that will be transported through the system? I have enabled "heavy ion physics" (#), tried options 3 and 5 for NCIA on the neutron...

I get 2^5 x 3^80 Am I correct?

I understand that dot product gives us a number and cross product gives a vector. Why is this vector orthogonal to the others two, and why it has magnitude |a|*|b|*sinΘ? How to use cross product? What does it give to us?

Summary:: Direction of reaction A = 2B and we know that there is 70% of products and K = 2.3 Question is: What is the direction of reaction A = 2B and we know that there is 70% of products. K = 2.3 K = B^2 / A B:A = sqrt (K.A) / A I have came here but what should I do now? Because when I...

Let $a,\,b,\,c$ be numbers such that $(a^2+b^2+c^2)\left(\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}\right)=26$ and $(a^3+b^3+c^3)\left(\dfrac{1}{a^3}+\dfrac{1}{b^3}+\dfrac{1}{c^3}\right)=78$. Find the value of $(a^4+b^4+c^4)\left(\dfrac{1}{a^4}+\dfrac{1}{b^4}+\dfrac{1}{c^4}\right)$ and...

Hey guys! I read this fascinating paper about the discovery of a white dwarf merger remnant: https://www.nature.com/articles/s41586-019-1216-1 To quote the abstract: "For white dwarfs, the mass of the merger product may exceed the Chandrasekhar limit, leading either to a thermonuclear explosion...

I don't understand why ? ## \Psi ∇ ⋅ (A \Psi^ *) + \Psi ^* ∇ ⋅ (A \Psi ) = 2 ∇ ⋅(A \Psi ^* \Psi) ## form ## ∇ ⋅ (fg) = ∇f ⋅ g + f(∇ ⋅ g) ## Attempt at a Solution ## \Psi ∇ ⋅ (A \Psi^ *) + \Psi ^* ∇ ⋅ (A \Psi ) = 2 ∇ ⋅ (A \Psi ^* \Psi) - ∇\Psi ^* ⋅ A\Psi - ∇\Psi ⋅ (A\Psi ^*) ##