Form Definition and 1000 Threads

Hello! (Wave)I found the following in my lecture notes:$$u_t=u_{xx}, x \in \mathbb{R}, t>0 \\ u(x,0)=f(x)$$ $$u(x,t)=X(x)T(t)$$ $$\Rightarrow \frac{T'(t)}{T(t)}=\frac{X''(x)}{X(x)}=-\lambda \in \mathbb{R}$$ $$X''(x)+\lambda X(x)=0, x \in \mathbb{R}$$ $$X \text{ bounded }$$ The characteristic...

Homework Statement Four charges equal in magnitude of 20.0 microC are placed on the four corners of a square with side length 0.180m. Determine the electric field at the centre of the square. (-q) ---------- (+q) l l l l l...

Homework Statement Find diagonal shape of next quadratic form ( using eigenvalues and eigenvectors) Q(x,y)= 5x2 + 2y2 + 4xy. What is curve { (x,y)∈ ℝ| Q(x,y)= λ1λ2, where λ1 and λ2 are eigenvalues of simetric matrix joined to quadratic form Q. Draw given curve in plane. The Attempt at a...

Hii! Newton's law of gravity is ∇.(∇Φ) = 4πGρ. A book on GR gives a suggestion to make it Lorentz covariant by using de' Alembertian operator on 'Φ' in the LHS of above equation instead of Laplacian. Then it explains that this won't work because we have to include in 'ρ' all the energy...

Homework Statement Problem: Consider a system of linear equations in echelon form with r equations and n unknowns. Prove the following.: (i) If r = n, then the system has a unique solution. (ii) If r < n, then we can arbitrarily assign values to the n - r free variables and solve uniquely...

Homework Statement Homework EquationsThe Attempt at a Solution Doing a vertical force balance 2Fcosθ=mg ,where m is the mass of water . Not sure how to proceed . What role does the pin hole at the top play ? I would be grateful if somebody could help me with the problem.

Hi, I want to transform a complex exponential with quadratic phase to discrete form, in other words to a vector form. can anyone help me with that? Thanks

I am currently designing a whole-core for small PWR. I am calculating core life (years) vs. FdH and RFF. What is the difference between "Enthalpy rise hot channel factor (FdH)" and "Radial form factor (RFF)"? I have calculated "CHANNEL FDH" which is same as "RFF". Do I have to try "CHANNEL...

1).Virtual Photons forms in pairs like particles and anti particles? or 2).by borrowing energy from future? or 3) in both ways?

Is it true that ##V(\textbf{r}) = \frac{1}{4 \pi \epsilon_0} \int \frac{1}{|\textbf{r}-\textbf{r}^{'}|}\ \rho(\textbf{r}^{'})\ d \tau^{'}## is the integral form of Poission equation ##\nabla^{2} V = - \frac{1}{\epsilon_0} \rho##? Can you trace the steps that lead from one to the other, or at...

I have completed the exercise, but I did something weird in one step to make it work, and I'd like to know more about what I did...or if what I did was at all valid. 1. Homework Statement Show that Laplace's equation \frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial...

So I've found several instances in which Bernoulli and Conservation of momentum (in Reynolds transport theorem form) give different answers for the study of an inviscid fluid. Let's consider a simple situation as described in my diagram attached. Basically a tap/faucet is releasing fluid, which...

In the GRW spontaneous collapse model (for example) the wave-function evolves by linear Schrödinger equation, except, at random times, wave-function experiences a jump of the form: \psi_t(x_1, x_2, ..., x_n) \rightarrow \frac{L_n(x)\psi_t(x_1, x_2, ..., x_n)}{||\psi_t(x_1, x_2, ..., x_n)||}...

For some reason I just can't seem to wrap my head around the idea of reducing a Matrix to row echelon form. I'm familiar with the steps that the textbooks and tutorials use and how it's done but when I try practicing on my own I feel lost. e.g. all I end up with are just a bunch of random...

At the exact center of a finite wire (i.e. a distance, say $L/2$ from each end), why can I not apply Gauss's Law in integral form to find an EXACT solution for the electric field? At the center of the wire, $E$ is entirely radial, so it seems like I should be able to draw an infinitesimally...

In paper Phys. Rev. B 29, 3153 – Published 15 March 1984 general potential form is introduced and from that form one can obtain different class of period potential V(u,r)=A(r)\frac{1+e\cos (2\pi u)}{[1+r^2+2r\cos (2\pi u)]^p} ##-1<r<1## , where ##r## is real number, ##m,p## are integers...

In most GR textbooks, the general form of a spherically symmetric metric is obtained by inspection which is acceptable. But in the textbook I'm reading, the author does that with a mathematical analysis just to illustrate the method. But I can't follow his calculations. In fact he omits much of...

I never quite understood slope-intercept form, my math teacher never really explained it too well. And so it kind of affects almost everything else I do. Like the scatter plots and lines of best fit sort of thing. And all the more advanced stuff I never understood when I was in advanced classes.

Let r:R2 →R3 be given by the formula Compute the second fundamental form with respect to this basis (Hint: There’s a shortcut to computing the unit normal n). I can't find thi shortcut, does anyone help me? I'm solving it with normal vector and first and second derivate, but I obtained...

Question: ##h_{t}+vh_{x}+v_{x}h=0## ##v_{t}+gh_{x}+vv_{x}=0## Write it in the form ##P_{t}+Q_{x}=0##, where ##P=(h,hv)^{T}##, where ##g## is a constant ##>0##, and ##v## and ##h## are functions of ##x## and ##t##. Attempt: I have ##Q=(vh,?)^{T}##, the first equation looks easy enough, but...

Homework Statement Let C be the straight line from the point r =^i to the point r = 2j - k Find a parametric form for C. And calculate the line integrals ∫cV*dr and ∫c*v x dr where v = xi-yk. and is a vector field Homework EquationsThe Attempt at a Solution For parametric form (1-t)i + (2*t)j...

I've been studying for my final exam, and came across this homework problem (that has already been solved, and graded.): "Show that the Galois group of ##f(x)=x^3-1## over ℚ, is cyclic of order 2." I had a question related to this problem, but not about this problem exactly. What follows is...

What is the proof for this $$ \int_a^b f(x) dx = 1/n\lim_{n\to\infty} (f(a) + f(a+h) + f(a+2h) +...+ f( a+ (n-1)h)) $$ h = (b-a)/n Also I think there is some summation form which can be converted to integral form how?

As far as I understand, ##I_{3}##, the component of isospin in a certain direction is additive, but ##I## is to be treated as a vector sum, is this correct? So, ##I_{3}=1/2## for ##u## quark, ##I_{3}=-1/2 ## for ##d## quark. Adding ##I_{3}## then for a proton we find ##I_{3}=1/2## and for a...

Homework Statement A long cylindrical wire of radius R0 lies in the z-axis and carries a current density given by: ##j(r)= j_0 \left( \frac{r}{R_0} \right)^2 \ \hat{z} \ for \ r< R_0## ##j(r) = 0 \ elsewhere## Use the differential form of Ampere's law to calculate the magnetic field B inside...

Hi guys, I have a few tries at this but I keep coming up wrong, so can anyone show me how to convert these two standard form equations into slope intercept form? 2x - 11y = 2, -6x + 3y = 9. Greatly appreciate any help offered :) Thank you!

Homework Statement data[/B] Solving differential k forms. Homework Equations I don't want to give any exact problems from my problem set. The Attempt at a Solution solution.[/B] The text I'm using, CH Edwards, is very abstract in this section and the explanation over a sped up, last class...

Basic question, I think, but I'm not sure. It is a step in a demonstration, so it would be nice if it were true. True or false? Why? If A is a real, symmetric, nonsingular matrix, then xTAx≠0 for x≠0.

Getting stuck on something I think that could be trivial. Maybe someone can see my mistake. consider the system: $x' = -2x + y$ and $y' = 2x - 3y$ a) Write the system in matrix form my solution $\overrightarrow{X} = (^x_y)$ so: $X' = (^{x'}_{y'})$ so $A = $ \begin{bmatrix} -2 & 1 \\ 2 & -3...

This is a long one if you have to follow all of my steps. If you are reasonably familiar with Killing forms then you can probably just skip to the three questions. Okay, I'm on my latest project which is to get some idea about how Dynkin diagrams and Coxeter labels work. (How do you pronounce...

Homework Statement Show that the Form factor is ##\frac{3(sin x - x cos x)}{x^3}##. Homework EquationsThe Attempt at a Solution [/B] I know that the form factor is simply the Fourier transform of the normalized charge density: F(q) = \int \frac{\rho}{Z} e^{-i (\Delta \vec k) \cdot \vec r}...

Homework Statement Is the reduced echelon form of a matrix unique? Justify your conclusion. Namely, suppose that by performing some row operations (not necessarily following any algorithm) we end up with a reduced echelon matrix. Do we always end up with the same matrix, or can we get different...

I've run into stumbling block here, is some sort of conversion required when your equation looks like: (x+1/2)^2 = 12(y - 3). Do I move it to standard form or can I graph it as is?

Hello Excuse me, but how do I sketch the phasor of a voltage that it's V=5cos(10t+30degrees) and how the V=5sin(10t+30degrees) ? I know that these can be converted as the R<angle polar form, with R being the Vmax, ie the 5, and the angle the phase. But what doesn't it matter if I have cos or...

Hi, Im trying to prove that a prime $p\neq 3$ is of the form $p=x^2 + 3y^2$ if $p \equiv 1 \pmod{3}$. I have think in a prove as follows: As we know that $-3$ is a quadratic residue mod p, we know that the ideal $(p)$ must divide $(x^2 + 3) = (x + \sqrt{-3})(x - \sqrt{-3})$ in the ring...

Homework Statement Show that sin(2θ) / (1 + cos(2θ) = tan(θ) - I've completed this part Hence find the value of cot(π/8) in the form a + b√(2), where a, b ∈ ℤ Homework Equations cot(θ) = (1 + cos(2θ)) / sin(2θ) The Attempt at a Solution I did the math, got (1 + cos(π/4)) / sin(π/4) = (1 +...

Homework Statement Hi there. This is not really a problem, I am only trying to understand something but I simply can't. So Rutherford scattering says that $$ \frac{d\sigma }{d\Omega}=(\frac{Ze^2m}{8\pi \varepsilon _0 p^2})^2\frac{1}{\sin ^4(\Theta/2)}|F(q)|^2$$ where $$F(q)=\int \rho (\vec...

okay so as usual I am stumped ( I am not sure if I dislike math, or simply those who proclaim to be teachers of it. Spouting off steps is not the same thing as teaching ) Anyway, I have a problem that requires me to determine the equation of the line described: Through (6,-4), perpendicular to...

I'm a sophomore Physics major currently taking Mechanics, and I recently noticed something when I was going over some homework. I am really good at Calculus, and I noticed I tend to do way better on the calculus based problems (i.e. work, finding force from potential energy etc) than some of the...

Homework Statement Given sin x = (eix - e-ix) / 2i, I want to prove that arcsin x = -i ln(ix + √1 - x2) Homework Equations I know about the Euler's formula and complex number. But I have never learned about complex logarithms. The Attempt at a Solution I try to use x = sin y. But it seems...

Just curious... I mean they don't form inside the event horizon, so how would they form outside? How does accretion do this?

Homework Statement Find the electric field inside and outside a sphere of radius R using the differential form of Gauss's law. Then find the electrostatic potential using Poisson's equation. Charge density of the sphere varies as ##\rho (r) = \alpha r^2 \ (r<R)## and ##\rho(r)=0 \...

Suppose some energy, say light energy transfers from one body to another .. Then will the light energy transferred to the second body also be in the form of light energy? I mean are there any chances that the energy transferred between two different bodies be of same kind of energy? If yes...

1. Homework Statement Three electrons form an equilateral triangle 1.00nm on each side. A proton is at the center of the triangle. What is the potential energy of this group of charges? Known Variables: s = 1.00 × 10-9m p+ Charge = 1.60 × 10-19C e- Charge = -1.60 × 10-19C r = s/√(3) = 5.77 ×...

I'm trying to show that any unitary matrix may be written in the form \begin{pmatrix}e^{i\alpha_1}\cos{\theta} & -e^{i\alpha_2}\sin{\theta}\\ e^{i\alpha_3}\sin{\theta} & e^{i\alpha_4}\cos{\theta}\end{pmatrix} Writing the general form of a unitary matrix as U=\begin{pmatrix} u_{11} & u_{12}\\...

Find a closed form expression for $$\sum_{k=1}^{n^2}\dfrac{n-\lfloor\sqrt{k-1}\rfloor}{\sqrt{k}+\sqrt{k+1}}$$.

When I deduct the the general form of wave equation, I noticed it has a second order partial derivative form. I am wondering why wave equation has a second order partial derivative form nor a first order partial derivative form?

Homework Statement Find all orthogonal 3x3 matrices of the form \begin{array}{cc} a & b & 0 \\ c & d & 1\\ e & f & 0 \\\end{array} Homework Equations There are many properties of an orthogonal matrix. The one I chose to use is: An n x n matrix is an orthogonal matrix IFF $$A^{T}A = I$$. That...

Homework Statement Here is my equation that I want to find theta angle Homework EquationsThe Attempt at a Solution I try to set different value of cos (theta) to find theta but it failed , I want to know the main solving strategy Thanks

Hello, In my problem I need to We are advised to create the Cooper pair box Hamiltonian in a matrix form in the charge basis for charge states from 0 to 5. Here is the Hamiltonian we are given H=E_C(n-n_g)^2 \left|n\right\rangle\left\langle...