Coefficients Definition and 805 Threads

Homework Statement Third question of the day because this assignment is driving me crazy: Suppose that \left\{ f_{k} \right\} ^{k=1}_{\infty} is a sequence of Riemann integrable functions on the interval [0, 1] such that \int ^{0}_{1} |f_{k}(x) - f(x)|dx \rightarrow 0 as k \rightarrow...

So, until now I know: (DV/DS)p=(DT/Dp)s=a*T/cp*(rho) (enthalpy) (Dp/DT)v=(DS/DV)t=-a/k (helmoltz) (DS/Dp)t=-(DV/DT)p=-Va (gibbs) a=expansion coefficient k=isothermal compression coefficent cp=heat capacity at constante pressure I want to deduce DT/DV at constant entropy=(DT/DV)s. BUT...

Hi, I have an issue with the Fresnel amplitude coefficients. I know they are given in two versions, for s-polarization and p-polarization. A version for unpolarized (randomly polarized) light is available for the power coefficients - it's just an average - but I could not find such an expression...

Hi, The Fourier series can (among others) expressed in terms of sines and cosines with coefficients a_n and b_n and solely by sines using amplitudes A_n and phase \phi_n. I want to express the latter using a_n and b_n. Using a_n = A_n \sin(\phi_n) \\ b_n = A_n \cos(\phi_n) I...

Homework Statement Consider 2 conductor spherical shells of radii a and b (where a>b). The inner shell is at zero potential and the outer shell is at a potential given by ##V(\theta, \phi )=V_0 \sin \theta \cos \phi ## where ##V_0## is constant and theta and phi are the usual spherical...

Hi everybody! I am working in a solid state physics group and need to find some heat transfer data to finish my calculations. I need to find the Nusselt number Nu=\frac{hL}{K} describing the heat transfer interface between a Si wafer and helium gas at a variety of temperatures. To find this I...

Why we always write equation in form y''(x)+a(x)y'(x)+b(x)=f(x) Why we never write: m(x)y''(x)+a(x)y'(x)+b(x)=f(x) Why we never write coefficient ##m(x)## for example?

Homework Statement I've reached a relation but then I need to obtain the coefficients ##A_m## and ##B_m##'s, those are the only unknowns. Here's the expression: ##\sum _{m=0}^\infty a^m [A_m \cos (m \theta ) + B_m \sin (m \theta )]=T_0\sin ^3 \theta##. Homework Equations Fourier...

Homework Statement Hello guys. I'm totally stuck at finding the solution to ##y'''-12y'+16y=32x-8##. Homework Equations Variation of parameters once I'm done with the general solution to the homogeneous ODE. The Attempt at a Solution First I want to solve the homogeneous ODE...

For the wave equation I managed to get the coefficient of f: a1=2 and the coefficient of g: \frac{12pi}{2pi*2}=B2 Is these answers right, since my B2 does not match the answer I was given. Thank you

I have to understand some papers concerning pzt actuated membranes for a project and I keep stumbling upon d31 or d33 strain coefficients. Can someone please explain to me what do the 31 and 33 numbers refer to, regarding the geometry? I'm not sure I'm getting this so please feel free to ask...

Homework Statement I must find the oscillations of a circular membrane (drum-like). 1)With the boundary condition that the membrane is fixed at r=a. 2)That the membrane is free. Homework Equations The wave equation \frac{\partial ^2 u }{\partial t^2 } - c^2 \triangle u =0...

Given the ODE of the form: y''(x) + A*y'(x) + B*y(x) = 0 If we choose a solution such that y(x) = e^{mx} and plug it into the original ODE, the ODE becomes: (m^{2} + A*m + B)e^{mx} = 0 If we solve for the roots of the characteristic equation such that m = r_{1}, r_{2} (root 1 and root 2...

$$ A\exp\left(\frac{-t}{2}\right)\sin x\cosh \left(\frac{\sqrt{5}}{2}t\right) + B\exp\left(\frac{-t}{2}\right)\sin x\sinh \left(\frac{\sqrt{5}}{2}t\right) $$ What would be the guessed form using the method of undetermined coefficients? Also, if I had $$...

Using the coefficients determined, combine the terms to arrive at the following approximation for the solution $$ u(r,\theta,\phi)\simeq 25 + \frac{75\sqrt{2}}{2}r\sin\theta\cos\phi + \frac{125}{\pi}r^2\sin^2\theta\cos 2\phi. $$ How do I combine them? The coefficients are For $\ell = 0$ and $m...

Homework Statement Given the function 10sin^2(10t) Find the fundamental frequency and period. Find the exponential and trigonometric coefficients of the Fourier Series. Homework Equations The Attempt at a Solution I really have no idea how to start this problem. The sin^2...

I just don't get why the method of undetermined coefficients can't be applied to tan(x) and sec(x). What my book says is this- "Since the number of terms applied by differentiating tan(x) and sec(x) is infinity". What do they mean by that? Even the number of terms obtained by...

find a, b, c, and d, such the cubic $f(x)=ax^3+bx^2+cx+d$ satisfies the given conditions Relative maximum (3,3) Relative minimum (5,1) Inflection point (4,2) I approached this by using the f'(x)= a(3)(x^2)+b(2)(x)+c with the min and max and f''(x)=6x+2b for inflection pt to get $27a +6b + c...

While reviewing for my midterm I came across an old problem that asked me to find the expansion coefficient for n=1 given an expression for the superposition wavefunction. I also know the expressions for the individual eigenstates because it is simply considering a particle in a 1-D box. I am...

Homework Statement Work out the correct coefficient arrays for these equations: y(n)=y(n-1)+\frac{1}{5}(x(n)-x(n-5)) y(n) = 0.82y(n -1) + .28x(n) Homework Equations \sum a(r)y(n+1-r)=\sum b(k)y(n+1-k) where a(1) = 1  The Attempt at a Solution Ok for the second equation...

Homework Statement Let f(x) = x^{4}+ax^{3}+bx^{2}+cx+d be a polynomial with real coefficients and real zeroes. If |f(i)| = 1, (where i = \sqrt{-1}) then find a+b+c+d. Homework Equations The Attempt at a Solution f(i) = 1-b+d+ci-ai Taking modulus |f(i)|= |1-b+d+i(c-a)|...

Anyone know how would I simplify a cross product where the two vectors have coefficients? For example (x/(y^3))\bar{r} X (x/(y))\bar{L} Thanks!

Homework Statement Part 1: I have a cylinder of radius R and lengh L. At first i can assume that we have expansion in both R and L. And that i can use the linear thermal expansion coefficient(\alpha) = 4 \times 10^{-6}. The relative change in R and L is 1 \times 10^{-4}, and from that i have...

Homework Statement Find coefficients A, B, and C. f'(x)= Af(x)+Bf(x+h)+Cf(x+2h)+O(h2) Using Taylor's Theorem. Note: O stands for Big O in asymptotic order notation. The Attempt at a Solution Here are the expansions: Bf(x+h)= Bf(x)+Bhf'(x)+(1/2)Bh2f"(x)+(1/6)Bh3f"'(x)...

Supposing $f$ is bounded and $A_n$ is given by 1-8, prove that $\sup_n|A_n|$ is finite. $$ f(\theta) = \sum_{n = -\infty}^{\infty}A_ne^{in\theta} $$ Since $f$ is bounded, $|f| < M = |z|\in\mathbb{C}$. Since it could be $\mathbb{C}$, $M$ would be the modulus correct? We know that the modulus of...

Homework Statement Expressing the binomial coefficients in terms of factorials and simplifying algebraically, show that (n over r) = (n-r+1)/r (n over r-1);Homework Equations The Attempt at a Solution I honestly don't even know how to come about this problem...I really need help in this...

Homework Statement Define (n k) = n!/k!(n-k)! for k=0,1,...,n. Part (b) Show that (n k) + (n k-1) = (n+1 k) for k=1,2,...n. Part (c) Prove the binomial theorem using mathematical induction and part (b). Homework Equations The Attempt at a Solution I'm wasn't able to...

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Folks, I am interested to know what the author is doing in the following ##\displaystyle B_{ij}=EL ij (L)^{(i+j-1)} \left[ \frac{(i-1)(j-1)}{i+j-3} -\frac{2(ij-1)}{i+j-2}+\frac{(i+1)(j+1)}{i+j-1}\right]## he states that this expression is not valid for ##B_{ij}## when ##i=1## and...

EDIT: I figured it out by looking at this link pages 65-66. Thanks for looking though! http://www.bfasta.net/assets/files/departments/science/ismith/Phys%20446/Phys%20446%20Information/HSU/Chapter%204%20Acceleration.pdf Homework Statement Recently I just did a physics lab for kinematics in...

Hiah, I've got a question concerning a t-pipe configuration and the corresponding friction coefficient values because there are two different friction coefficients stated in literature. Let's assume we have a simple t-pipe where the main passage is larger than the side branch. The friction...

I'm having trouble proving the following identity (I don't even know if it's true): $$\sum_{r=1}^k \binom{k}{r} \binom{n-k-1}{r-1}=\binom{n-1}{k-1}$$ $$\forall n,k \in \mathbb{N} : n>k$$ Thank you in advance for any help! Vincent

I have a quick question. The problem reads: Prove that there is no integer m such that 3x2+4x + m is a factor of 6x4+50 in Z[x]. Now, Z[x] is not a field. So, the division algorithm for polynomials does not guarantee us a quotient and remainder. When I tried dividing 6x4+50 by 3x2+4x +...

Homework Statement I found everything except step #5. Please tell me if I am correct Find a particular solution to (D - 1)(D^{2} + 4D - 12)y = cos(t) using the annihilator approach of the method of undetermined coefficients. Homework Equations 1) Find annihilator 2) Find A =...

I'm trying to estimate the heat transfer coefficient from the surface of a hot object moving through open air. After much searching, all I can find is coefficients for sundry fluids confined within the tubes of heat exchangers. I'm trying to calculate for a body at between 300-460K moving at...

Homework Statement I am working on this ahead of my fall class and don't actually want the answer... just pointers to help me understand something.. Thanks guys! :) I am really rusty with my general physics and calculus knowledge =( The original question asks me to prove that, for a...

Homework Statement http://gyazo.com/6c440aa92106f729639c91f6d59dcd89 The Attempt at a Solution My question is why is yp = At^3+Bt^2 +Ct. The reason I ask that is because is see t^2+2t so why wouldn't it be yp=At^2 +Bt +c?

Homework Statement Calculate the Einstein coefficient of spontaneous emission for nitrogen transitioning from Ei: 32P1/2 to Ek: 32S1/2. Total duration τ = 16ns, emission wavelength λ = 589.593 nm. Homework Equations The Attempt at a Solution I want to use A_{ik} =...

I just want to know if I'm understanding this right. I haven't really seen homology/cohomology outside of Z-related coefficients before, so this still seems kind of weird. I also haven't actually learned sheaf theory, so this might just be totally wrong. So if I have a top space and a sheaf...

Hello! I have some examples of non-homogeneous ODEs to be solved by the undetermined coefficients method. Two from "Pauls math notes" page: y''+8y'+16y=e^{-4t}+(t^2+5)e^{-4t} The compsol. for this is: Y_{c}=C_{1}e^{-4t}+C_{2}te^{-4t} The first guess for a particular solution would be...

Suppose you look at polynomials, P(x), of degree n, with all nonzero integer coefficients and, in particular, a coefficient of 1 for the nth degree (leading) term. And look at those polynomials whose squares have the fewest number of nonzero integer coefficients possible. Examples...

Homework Statement For any quadratic polynomial ax2+bx+c having zeros β and α Prove that β + α = -b/a and αβ = c/a. Homework Equations The Attempt at a Solution I have found a method myself to prove α+ β = -b/a. However, I could not prove αβ = c/a. It goes like this. If α and β are the...

I have a Mathematics C assignment, with one question being about static and dynamic friction. But I think it fits this forum. Anyway, we need to conduct experiments to show if there is a difference between static and dynamic friction or not with three different surfaces.The weight of the object...

A polynomial $f(x)$ has Integer Coefficients such that $f(0)$ and $f(1)$ are both odd numbers. prove that $f(x) = 0$ has no Integer solution

Hello. I'm having some trouble balancing ionic equations.. Are we supposed to consider the coefficients of the reactants/products? I came across a contradiction in the examples given in my book : 1) Na + H(+1) → Na(+1) + H2 So in order to figure out the oxidation/reduction part, we...

Let a,b,c,d be real numbers. Sauppose that all the roots of the equation $z^4 + az^3 + bz^2 + cz + d = 0$ are complex numbers lying on the circle $\mid z\mid = 1$ in the complex plane. The sum of the reciprocals of the roots is necessarily: options a) a b) b c) -c d) d ---------- Post...

Homework Statement Solve the following equation: (2x-y)dx+(4x+y-6)dy=0 Homework Equations Solve for M and N as a linear system of equations; and x_t = u + h y_t = v + k The Attempt at a Solution M = 2x-y=0 N = 4x+y-6=0 2x=y 4x+2x=6 6x=6 x=1 y=2 ∴ x_t=u+1 \\ dx=du...

Homework Statement If C^1(\mathbb T) denotes the space of continuously differentiable functions on the circle and f \in C^1(\mathbb T) show that \sum_{n\in\mathbb Z} n^2 |\hat f(n)|^2 < \infty where \hat f(n) is the Fourier coefficient of f. The Attempt at a Solution Since f is...

Homework Statement I have some past exam questions that I am confused with Homework Equations a_{n} = \frac{1}{2\pi i} \oint_\gamma \frac{f(z)}{z-a}\, dz The Attempt at a Solution I'm not sure how to approach this, I'm completely lost and just attempted to solve a few: a) it says f(z)...

Homework Statement Find a suitable form for the general solution of y'' - 4y' + 4y = 2t2 + 4t*e2t + t*sin(2t) For respective particular solutions, state where it is a general or a special case. DO NOT evaluate coefficents.Homework Equations Y1 = At2 + Bt + C Y2 = (Dt3 + Et2) * e2t Y3 = (Ft +...