Form Definition and 1000 Threads

Hey! How does the operator of angular momentum operates in exponential form? $$ e^{-i\theta J}\vert l, m \rangle = ?? $$ where $$J\vert \Psi \rangle = J\vert l, m \rangle$$ and $$J^2\vert \Psi\rangle = \hbar^2 l(l+1)\vert \Psi\rangle $$ Also, how do you operate $$J_-$$ and $$J_+$$...

Hi all, I am a final year maths student and am doing my dissertation in the finite element method. I have gotten a little stuck with some parts though. I have the weak form as a(u,v)=l(v) where: $$a(u,v)=\int_{\Omega}(\bigtriangledown u \cdot\bigtriangledown v)$$ and $$l(v)=\int_\Omega...

When there is a water wave in a ripple tank( not involving any interference experiment ) there will be a pattern on the bottom of the ripple tank , that is the the dark and bright fringes. How does the bright fringes and dark fringes form for only water waves experiment in ripple tanks ? It...

I am trying to prove the following. Let $V_1, \ldots, V_k$ be finite dimensional vector spaces over a field $F$. There is a natural isomorphism between $V_1^*\otimes\cdots\otimes V_k^*$ and $\mathcal L^k(V_1, \ldots, V_k;\ F)$. Define a map $A:V_1^*\times\cdots\times V_k^*\to \mathcal L^k(V_1...

I know the solution has an infinite number of solutions. It is represented as follows: x1= 4/3 + (1/3)x3 - (5/3)x4 x2= 2 + (1/3)x3 + (1/3)x4 x3= Free x4= Free How do I put the above solution into vector form as illustrated in the original question?

So in deriving the metric, the space-time can be foliated by homogenous and isotropic spacelike slices. And the metric must take the form: ##ds^{2}=-dt^{2}+a^{2}(t)\gamma_{ij}(u)du^{i}du^{j}##, where ## \gamma_{ij} ## is the metric of a spacelike slice at a constant t QUESTION: So I've read...

I was taught that when you have a polynomial fraction where the denominator is of a higher degree than the numerator, it can't be reduced any further. This seems wrong to me for a couple of reasons. 1. If the denominator can be factored some of the terms may cancel out 2. Say you have the...

Find the solution of the given initial value problem in explicit form. Determine interval which solution is defined. (which i think is the same thing as saying find the interval of validity) $y' = (1-2x)y^2$ , $y(0) = -1/6$ So here is what I have so far.. $\int y^{-2}dy = x - x^2 + C$ $=...

When dealing with this separable equation for example, if I'm told to solve the given D.E. $y' = x^2/y$ so after manipulation and taking the integral I got $\frac{y^2}{2} = \frac{x^3}{3} + C$ This is the implicit form correct? Would the explicit form be $y = \sqrt{\frac{2}{3} x^3 + C}$

Homework Statement Hi, I need help in solving question c) (a pendulum) The required data, problem and relevant equation is in the pictureThe Attempt at a Solution I am not sure how to solve it but here are my thoughts: since mg is working at j y(t)j= mg does that mean K(r-L0) x(t) direction? I...

I'm looking at deriving the Schwarzschild metric in 'Lecture Notes on General Relativity, Sean M. Carroll, 1997' and the comment under eq. 7.8, where he seeks a diagnoal form of the metric... - Is it always possible to put a metric in diagonal form or are certain symmetries required? - What...

Solve the IVP. $(2x-y)dx + (2y-x)dy = 0 $. $y(1) = 3$. Leave solution in implicit form. So I got: $\frac{dy}{dx} = \frac{-(2x-y)}{2y-x}$ Would this be correct since I didn't explicitly solve for $dy$ ?

Hello everyone, I'm a young alpine skier and looking for ways to be faster. Slalom guards haven't changed in form since over a decade! I want to make a shin guard that could decrease the impact of the slalom gate on your shins so it doesn't make you slower. I added some videos in slow motion...

I have the weak form of the poisson equation as $a(u,v)=l(v)$ where $$a(u,v)=\int_\Omega \bigtriangledown u\cdot\bigtriangledown v$$ I have been proving existence and uniqueness of the solution using the Lax-Milgram Thm. I am stuck on proving that the bilinear form $$a(u,v)$$ is coercive. I...

Homework Statement What is the amplitude and phase of the complex function? f(t) = (1-2i)e^(iwt) Homework Equations None/unknown Normal Polar Form = Real*e^imaginary i = e^pi/2*i The Attempt at a Solution [/B]I am trying to bring this into a normal polar form to easily see the phase and...

I mainly just need some clarification here. I was doing my homework and then browsing the web to find an answer to my problem and came across mathewarehouse' definition of Standard form and then I looked at my homework and went..."huh?" I don't understand if my homework is listening this wrong...

i just wanted to get this cleared that a beam falling on a diffraction grating with a shape gives the Fourier images of the grating object which can be reobtained by placing a biconvex lens that would converge the rays and form a focussed Fourier image at its focal length and the image of the...

sinh(ln2+ ipi/3) so I have a general formula of sinh z= (e^z - e^-z)/2 so I obtained the following sinh= (e^(ln2 + ipi/3) - e^-(ln2 + ipi/3))/2 sinh= (e^(ln2) e^(ipi/3) - e^(-ln2) e^(ipi/3))/2 e^(ln2)=2 e^(-ln2)=1/2 e^(ipi/3)= (1/2)+(isqrt3/2) e^(-ipi/3)= (1/2)-(isqrt3/2) so when I plug...

Homework Statement Write the following statements in predicate form, using logical operators ^,∨, (NOT - negation but don't know where the symbol is :/) , and quantifiers ∀,∃. Below ℤ+ denotes all positive integers {1,2,3,...}. I need help with this first statement: For any x, y ∈ ℤ+ the...

Homework Statement ((1-i)/(sqrt2))^42 express in x+iy form Homework Equations z1/z1=(r1/r2)e^(i(theta1-theta2)) The Attempt at a Solution Ive found that (1-i) has r=sqrt2 so since r is sqrt2 and x=1 y=-1 so the angle is 7pi/4 so then I have (sqrt2e^(-i7pi/4)/sqrt2)^42 now from here is where I...

I have the feeling that it is, but I am not really sure how to start the proof. I know I have to prove both closure axioms; u,v ∈ W, u+v ∈ W and k∈ℝ and u∈W then ku ∈ W. Do I just pick a vector arbitrarily say a vector v = (x,y,z) and go from there?

Greetings. I have been teaching myself Calculus. To do this I ordered a used Larson's 8th Edition Calculus and a used TI-81 graphing calculator. When I got to Chapter 10, I ran into a problem: the chapter introduces equations in polar form and when I whipped out my TI-81, I had no idea how to...

What is the domain of $$y^2-2y=x^2-x-1$$? I don't know how to find it for implicit functions.

Conventional GR is based on the Levi-Civita connection. From the fundamental theorem of Riemann geometry - that the metric tensor is covariantly constant, subject to the metric being symmetric, non-degenerate, and differential, and the connection associated is unique and torsion-free - the...

Write the equation for an ellipse with vertices (0,-3) and (0,3), minor axis of length 10. I know that the standard form of an ellipse is (x-h)^2/a^2 + (y-k)^2/b^2=1 Please help me ! Please! Thank you so much for your time, I appreciate it.

Homework Statement A molecule has a velocity v and speed v. I've worked out (and understand) that the number of molecules in a gas with speeds between v and v+dv and moving at angles between Ө and Ө+dӨ to any chosen axis is: (1/2)nf(v)dvsin(Ө)dӨ The internet verifies this. f(v) is the speed...

I'm looking for a proof of the following statement at a level an early undergrad would understand: ##J=D \vec{\nabla} \vec{n}## where ##D=\frac{v_{th}l}{3}## with ##l## being the mean free path and ##v_{th}## the thermal agitation velocity, ##J## is the particle current density. I really did...

Please check my solution.View image: Possible Echelon Form of Matrix

Please Check My Solution View image: Possible Echelon Form of Matrix

Hi! (Smile) Suppose that we index the components of the elements of $\mathbb{Z}_p$ by subscripts. Indexing the terms of the sequence by superscripts in parentheses$x^{(i)}$ is a term of the sequence, and $x^{(i)}_k$ its $k$-th component. So, if we have a sequence in $\mathbb{Z}_p$, it will be...

Homework Statement Why is it necessarily true that for a hyperbola, the focus length, ##f ## has got to be greater than the semi-major axis , ## a## - ## f >a ## ? Homework Equations - The Attempt at a Solution I needed to derive the cartesian equation of a hyperbola with centre at ##...

When light enters some pieces of glass from the air, such as a magnifying glass or window, rainbows usually don't form. But when light enters a prism, rainbows form. Why do rainbows form in the prism, but not in the magnifying glass or window?(This is my own personal curiosity and because I...

why $$1^\infty$$ is indeterminate form?

I just finished a course on linear algebra which ended with Jordan Canonical Forms. There were many statements like "Jordan canonical forms are extremely useful," etc. However, we only learned a process to put things into Jordan canonical form, and that was it. What makes Jordan canonical...

While fiddling around with some very simple linear ODEs, I "discovered" a formula that gives a solution to ODEs of the form: ##y'+y=ax^n ##. here it is: i'm sure that this was discovered before, but i was just wondering if it had any official name or something.

Context: Deriving the maximally symmetric- isotropic and homogenous- spatial metric I've seen a fair few sources state that the Rienamm tensor associated with the metric should take the form: * ##R_{abcd}=K(g_{ac}g_{bd}-g_{ad}g_{bc})## The arguing being that a maximally symmetric space has...

If we have a vector $$\partial_v$$ and we want o find its norm, we easily say (According to the given metric) that the norm of that vector is:$$ g^{vv}\partial_v\partial_v$$. My question what if we have a vector that is combination of 2 vectors like: $$\phi =\partial_v + a\partial_x$$ where $a$...

I was solving this problem and I didn't want to do it the really long way by finding the equation of B(t) by first finding T(t) and N(t). So i took the cross product of r' and r'' so that they would be in the direction of B. Found the parametric equation of the plane but the book answer was in...

What is the state of carbon just prior to it forming a diamond, deep in the Earth's crust ? Carbon vapour , or liquid carbon ?.Update: I think I've managed to answer my own question ... http://www-als.lbl.gov/images/stories/Science_Highlights/Highlights/108carbon1.png...

Homework Statement Find the derivative of f(X). f(X) = transpose(a) * X * b where: X is nxn a and b are n x 1 ai is the i'th element of a Xnm is the element in row n and column m let transpose(a) = aT let transpose(b) = bT Homework Equations I tried using the product rule...

Hi! I'm studying physics and currently taking the first mechanics course. After dealing with rotation and gyroscopes, now we're working on things like shear stress, and Hooke's law in tensor form etc. I've got Kleppner/Kolenkow but shear stress, Hooke's law in tensor form and tensors in...

Homework Statement Sorry if this is a dumb question, but say you have 1/(1-x) This is the form of the geometric series, and is simply, sum of, from n = 0 to infiniti, X^n. I am also trying to think in terms of Binomial Series (i.e. 1 + px + p(p-1)x/2!...p(p-1)(p-2)(p-(n-1) / n!). 1/(1-x) is...

I have a question why everyone says ∫uv' dx=uv-∫u'v dx why don't they replace v' with v and v with ∫vdx and say ∫uv dx=u∫vdx-∫(u'∫vdx) dx i think this form is a lot simpler because you can just plug in and calculate, the other form forces you to think backwards and is unnecessarily complicated.

I was tutoring a student and I came across the following question. I feel like I'm missing something obvious, but it seems like there are too many variables for an answer to be determined. The attached picture contains all of the question details.

I have to show the roots of $$x^{2}-8x-29=0$$ are c$$\pm$$d$$\sqrt{5}$$ I used completing the square method. Once I used CTS I got the answer $$(x-4)^2-45=0$$ So I am not sure what is the next step to put it in the form of c$$\pm$$d$$\sqrt{5}$$

I am trying to prove the following standard result:Let $V$ be a finite dimensional vector space over a field $F$ and $f:V\times V\to F$ be a symmetric bilinear form on $V$. Let $W$ be a subspace of $V$ such that $f$ is non-degenerate on $W$. Then $$V=W\oplus W^\perp$$(Here $W^\perp=\{v\in...

Homework Statement I am looking for some quick methods to integrate while leaving each step in its vector form without drilling down into component-wise integration, and I am wondering whether it is possible here. Suppose I have a square domain over which I am integrating two functions w and...

Homework Statement Find the following Fourier series in trigonometric form. Homework Equations $$y(t)=a_0+\sum\limits_{n=1}^{\infty} a_n cos(n\omega_{0}t)+b_n sin(n\omega_{0}t)$$ The Attempt at a Solution The graph above is represented by the function: $$ x(t) = \left\{ \begin{array}{ll}...

Homework Statement Find a closed form expression for the function f(x) which the power series Σn=0..∞ n(-1)nxn+1 converges to and determine the values of x for which f(x) equals the given power series. Homework Equations N/A The Attempt at a Solution I'm actually not sure how to start. First...

Dear all, please see the page above, (Alan F, Beardon, Abstract Algebra and Geometry). On the page, Theorem 3.5.2 says that the set of Complex numbers from ## z^n = 1 ##, where ## |z| = 1 ## forms a group w.r.t multiplication. I want to know if... The inverse of all elements...