Expression Definition and 999 Threads

Homework Statement Without explicit calculation, argue why the following expression cannot be correct: $$\nabla \times (\mathbf{c} \times \mathbf{r}) = c_{2}\mathbf{e_{1}}+c_{1}\mathbf{e_{2}}+c_{3}\mathbf{e_{3}}$$ where ##\mathbf{c}## is a constant vector and ##\mathbf{r}## is the position...

I have a large quantity N, which starts off equal to a determinable value N0. Over a short time ∆t, the value of N changes by -∆t*(B*N - C) where B and C are determinable constants. Am I correct in thinking I can turn this into: dN/dt = -(B*N - C) How do I get this into a formula for N at...

Homework Statement I'm having a bit of trouble with this problem: "A spherical ball of charge has radius R and total charge Q. The electric field strength inside the ball (r≤R) is E(r)=r^4Emax/R^4. a. What is Emax in terms of Q and R? b. Find an expression for the volume charge density ρ(r)...

Ian in London (South).Interests Number Theory. Prime Numbers... Integer iterations of this quadratic expression only yield decimal numbers whose factors end with the digit _1, or _9. Why?

Homework Statement I am trying to derive an expression for the potential of a positive point charge by bringing in another positive test charge in from infinity to a point at a distance R from the point charge. Homework Equations $$V_f - V_i = - \int \vec E \cdot d \vec r \, dr$$ The Attempt...

The cost of running a machine is partly constant and partly varies as the number of parts machined. Write an expression to show the cost? My Reasoning: Let C = cost Let x = partly constant Let y = partly varies Let k = constant of proportionality C = x + yk Right?

In a proof, I encountered the following expressions: \[\sum_{cyc}\frac{\cos^2 A}{\sin B \sin C}\geq \sum_{cyc}\frac{\cos B \cos C}{\sin B \sin C}=\sum_{cyc}\cot B \cot C =1\] My question is concerned with the validity of the inequality. The inequality is based on the use of the Rearrangement...

The question is about to derive an approximate expression by regular solution theory, It is difficult for me to find relevant source on this question. However, the question to me is so vague that I do not know how to answer. What I have tried is to search what the interaction parameter is...

The conservation of probability says: $$\partial_t J^{0} + \partial{i}J^{i} = 0$$ Use the Schrodinger equation to obtain$$ J^{i} (\vec r)$$. I have no idea where to start this kind of problem because the notation makes no sense to me. I would appreciate a hint or nudge in the correct direction.

\sqrt{3} + \sqrt{2} / \sqrt{3} - \sqrt{2}

< Mentor Note -- thread moved to HH from the technical forums, so no HH Template is shown > Hi all, I'm trying to find an expression for the current i(t) as a function of C and V_out (t) after t = 0. I have an answer but still not sure at all if its correct or not. Here is my answer: I know...

Homework Statement There is a thin rod with charge Q that has been bent into a semicircle with radius R. Find an expression for the electric potential at the center. Radius = R Charge = QHomework Equations V = ∫(k * dq)/rThe Attempt at a Solution dq = λdx λ = Q/L L = pi * r V = kQ / LR ∫ dx =...

if ## a,b,c,d,e ## are positive real numbers, minimum value of (a+b+c+d+e)( \frac{1}{a} +\frac{1}{b} +\frac{1}{c} +\frac{1}{d} +\frac{1}{e} ) (A) 25 (B) 5 (C) 125 (D) cannot be determined My approach : expanding the expression , i get 5+a( \frac{1}{b} +\frac{1}{c} +\frac{1}{d} +\frac{1}{e}...

Based on the vast cosmology texts, I have seen that the expression for the tensor-to-scalar ratio ##r## in cold inflation is, ##r=16\epsilon_H = -16\frac{\dot H}{H^2}\qquad## where, ##~~~\epsilon_H = -\frac{\dot H}{H^2}## ##H## is the Hubble parameter, and ##\epsilon_H## is the Hubble slow...

Homework Statement Hi guys, would just too make sure my derivation and insight to why is correct. Question: a) only Homework EquationsThe Attempt at a Solution $$dU=-wd_{ext}$$ $$dU=-F_{ext} \cdot dx$$ Now as the ##F_{ext}## is in the same direction and the direction vector { have not...

Hi, I've got this: $$\sin{(A*B)}\approx \frac{Si(B^2)-Si(A^2)}{2(\ln{B}-ln{A})}$$, whenever the RHS is defined and B is close to A ( I don't know how close). Here ##Si(x)## is the integral of ##\frac{\sin{x}}{x}## But, to check it, I need to evaluate the ##Si(x)## function. I'm new with Taylor...

$\tiny{Q5|8.5.17}$ $\textsf{Evaluate}$ \begin{align*}\displaystyle I&=\int_0^{9}\frac{x^3 \, dx}{x^2+18x+81} \color{red}{=243\ln2-162} \end{align*} OK even before I try to get this answer trying to see what road to take u substituion long division and remainder partial fractions (the...

Homework Statement The assignment is to find a closed-form expression for the FIR least squares inverse filter of length N for each of the following systens Homework Equations $$ 1.G( z ) = \frac{1}{1 - \alpha z^{-1}}; | \alpha | < 1 \\ 2. G(z) = 1 - z^{-1} \\ 3. G(z) = \frac{\alpha -...

Homework Statement A thick, spherical shell of inner radius a and outer radius b carries a uniform volume charge density ρ. Find an expression for the electric field strength in the region a<r<b. Homework Equations Gauss's Law ∫E⋅dA Area of a sphere The Attempt at a Solution I know I am...

$\tiny{242t.08.02.09}$ $\textsf{ Evaluate to nearest thousandth}\\$ \begin{align*} \displaystyle I_{8.1.31} &=\int_5^{10} \frac{3x^5}{x^3-5} \, dx =885.576\\ \end{align*} $\textit{ok tried getting to this answer by: }$ $u=x^3-5 \therefore du=3x^2 \, dx$ $\textit{but it didnt seem to go to good}$

I'm reading a book that mentions writing an algebraic expression in terms of its prime factors, for example: x2 - 2 x - 3 = (x + 1) (x - 3) I know what 'prime factors' means for a number but not for an expression. Aren't these just 'factors'?

Homework Statement In the diagram, Y006 is a lamp, and X & R references are relay contacts. Write down a Boolean expression for the lamp to light. Homework EquationsThe Attempt at a Solution This is what I have done as an interim step to help me visualise what's going on: This is the...

Homework Statement Given that ##\sinh x = \frac{e^x-e^{-x}}{2}##, find an expression for ##arcsinh x## Homework EquationsThe Attempt at a Solution We can proceed by the normal procedure for finding inverses of well-defined functions, solve ##x = \frac{e^y - e^{-y}}{2}## for y. After doing some...

$\tiny{206.07.05.88}$ \begin{align*} \displaystyle I_{88}&=\int\frac{1}{(x+2)\sqrt{x^2+4x+3}} \, dx \\ &=\int\frac{1}{(x+2)\sqrt{x^2+4x+3+1-1}} \, dx \\ &=\int\frac{1}{(x+2)\sqrt{(x+2)^2-1}} \, dx \\ u&=(x+2) \therefore du=dx \\ I_{88}&=\int\frac{1}{u\sqrt{u^2+1}} du \end{align*} $\textit{so...

There have been occasionally questions/comments about gene control. This should feed the hunger for knowledge. Here is an open access review article: Three-Dimensional Genome Organization and Function in Drosophila. It discusses some not widely known aspects of control of gene expression in...

Homework Statement $$\lim_{x\to\infty} \left(\frac{n^2+2n+1}{n^2+2n+3}\right)^{\frac{2n^2}{n+1}}$$ Homework Equations 3. The Attempt at a Solution [/B] I tried ##\lim_{x\to\infty} \left(\frac{n^2+2n+3-2}{n^2+2n+3}\right)^{\frac{2n^2}{n+1}}=## ##\lim_{x\to\infty}...

Basically i need to evaluate the limit of this expression ##\lim \sqrt[3]{n^6-n^4+5}-n^2=?## I want to know if this is correct and why: ##\lim \sqrt[3]{n^6-n^4+5}-n^2=\lim...

Homework Statement In an experiment, a student wishes to use a spring to accelerate a cart along a horizontal level track. The spring is attached to the left end of the track and produces a non-linear force of magnitude Fs = Ass + Bs, where s is the distance the spring is compressed in meters...

If anyone value in an expression is approaching 0, does the entire expression equal 0? So for example, for the limit $\lim_{{z}\to{q}} z ln(z) f(z)$. If $\lim_{{z}\to{q}}f(z) = 0$, then does $\lim_{{z}\to{q}} z ln(z) f(z)$ equal 0?

I don't quite get this question, how is it done ?

Homework Statement See picture above Homework Equations Int (E dot dl) = Int (B dot dA) The Attempt at a Solution Where did I go wrong? Is there a sign error possibly or am I off conceptually? [/B]

We were writting a test on wednessday and for the first question we were asked to calculate the angles of the base of a tetrahedron with points A,B,C,D. ABC being its base. We were also given some relation which helped me calculate that the base of the tetrahedron is an equilateral triangle...

Hi, I have ## 120 \sum \limits_1^\infty (\sigma_{3}(n))^{2} ## , where ## \sigma_{3}(n) ## is a divisor function. And I want to show that this can be written as ##120\sum \limits_{k=1}^{n-1} \sigma_{3}(k) \sigma_{3}(n-k) ## I'm pretty stuck on ideas starting of to be honest, since the sum is...

Having a big issue working through this problem and was wondering if somebody, much smarter than I, could give me a few pointers. Anyway here's the problem: Given that k2=εμω2-iωμσ By writing the wave number as k=kr-iki. Show that ki, which determines attenuation, can be expressed by...

Homework Statement Find the net force of water on the dam The dam has a width of w and the water is at a depth of d. So my question is, would you have to integrate ρgwd with respect to d? because the pressure is constantly changing every value of d. Homework Equations ρgwd ρ is the density of...

Homework Statement Find critical numbers of the function: F(x)=t^3/4 - 2t^1/4 Derivative I got: F'(x)=3/4 t^-1/4 - 1/2 t^-3/4 Homework EquationsThe Attempt at a Solution I have found the derivative and I understand I must pull out a t in order to find critical numbers, and run across this...

Let's say that we have this expression, $\frac{1}{2}*2x*x-2$ So now solving this in one method we can $\frac{1}{\cancel 2}*\cancel2x*x-2=x^2-2$ In another way $\frac{1}{2}*2x*x-2=\frac{2x^2-2}{2}=x^2-1$ What is the reason that I get two different answers ? I hope that both ways are valid...

I know that different cells in a body express different genes and thus have different functionalities. I am wondering if there are genes that are expressed in all the cells of a certain organism. (or for that matter in all the cells that have that specific gene, for example all mammals.)? For...

Hello. Given a range of time in which an event can occur an indefinite number of times, we say a random variable X folows a poisson distribution when it follows this statements: X is the number of times an event occurs in an interval and X can take values 0, 1, 2, … The occurrence of one event...

I was wondering whether this expression could be factored by any means $x^2+4x-1$Many THanks :)

I am trying to derive an expression for the Tension T in the massless bar in the given photo. Where there is a cart that is moving only in the x-direction and the bar is rotating around a point pivoted at the cart with angle theta. The expression that I have is deduced from: ∑ F (towards...

Homework Statement Two rods of the same material, of equal length l, and of cross section A and 2A respectively, are mounted between two rigid (i.e., nondeformable) cross frames. The frames are pulled by a pair of forces P, located at distance x from the thin bar. Derive an expression for the...

The problem I have been trying to solve a long problem but my answer differs from my books answer with just a few peculiar terms. My answer: ##x_1' \vee x_0'x_1x_2 \vee x_0x_1'x_2' \vee x_2## Book: ##x_1' \vee x_2 ## My question is: Is it possible to simplify ##x_0'x_1x_2 \vee...

Homework Statement The coefficient, β, of thermal expansion of a liquid relates the change in the volume V (in m3) of a fixed quantity of a liquid to an increase in its temperature T (in °C): dV = βV dT (a) Let ρ be the density (in kg/m3) of water as a function of temperature. (For a mass m of...

Homework Statement (a) A body of mass m is suspended from a vertical, light, helical spring of force constant k, as in Fig. 1. Write down an expression for the period T of vertical oscillations of m. (b) Two such identical springs are now joined as in Fig. 2 and support the same mass m. In...

Hello to everybody, I am solving some examples related to wave equation of shear horizontal wave in cylindrical coordinates (J.L Rose: Ultrasolic Waves in Solid Media, chapter 6), which is expressed as follows: ∇2u=1/cT2⋅∂2u/∂t2 The Laplace operator in cylindrical coordinates can can be...

I'm really lost on this one. I'm really not sure how to get started on this. I started it out as a force problem and solved for ax = g*cos(theta). I then integrated that (treating it like a constant) from zero to D to find my initial velocity at the time the block reaches the edge of the...

Homework Statement Given the diagram below, showing the path of a geocentric satellite S flying over a ground station G, find an expression for the geocentric semi-angle ##\phi## in terms of ##\epsilon##, the radius of the Earth ##R_E##, and the height of the orbit ##h##. Homework Equations...

Data From the rectangular glass sheet ABCD the isosceles triangular part ADE is cut away (See figure) The length of CE is 1m. Problem i. Take the length of DE as x meters, write an expression in terms of x , for the area of the remaining part of the sheet. The area of the remaining part...

I am attempting to understand how I can derive a new expression as a function of another variable in this existing expression. This is a college undergraduate course but I can't recall doing this much before, maybe in high school. I have taken all the way up to ODE's and PDE's. The confusion...