Partial Definition and 1000 Threads

Homework Statement divide in partial fractions: x^3 -3x^2+x-12 / x^4+5x^2+4 Homework EquationsThe Attempt at a Solution I factored x^4+5x^2+4 to (x^2 +1) (x^2+4) x^3 -3x^2+x-12/(x^2 +1) (x^2+4) = A (x^2+1)/(x^2 +1) (x^2+4) + B (x^2+4) /(x^2 +1) (x^2+4) they all have the same...

How can I figure out ##\partial_\mu x^2## on the manifold ##(M,g)##? I thought that it should be ##2x_\mu##, but I think I'm wrong and the answer is ##2x_\mu+x^\nu x^\lambda \partial_\mu g_{\nu\lambda}##, right?! In particular, it seems to me, we can't write...

Ok so I took partial fraction decomposition in Calc II, and now I'm taking it again in Differential Equations course. The problem is that I don't really understand what I'm doing. I understand the procedure when having simple real roots, for example 2x+1/(x+1)(x+2), it becomes A/(x+1) + B/(x+2)...

Homework Statement Find ∂2f ∂x2 , ∂2f ∂y2 , ∂2f ∂x∂y , and ∂2f ∂y∂x . f(x, y) = 1,000 + 4x − 5y Homework EquationsThe Attempt at a Solution Made somewhat of an attempt at the first one and got 0, however my teacher has poorly covered this in class, and I would value some further explanation.

Homework Statement A determinant a is defined in the following manner ar * Ak = Σns=1 ars Aks = δkr a , where a=det(aij), ar , Ak , are rows of the coefficient matrix and cofactor matrix respectively. The second term in the equation is the expansion over the columns of both matrices, δkr is...

1. The problem/question is as follows: 1 mole of O2 mixed with N2 gas (PN2= 5 atm at 10 degrees celcius in a 1 L flask. What is the total pressure after 2 moles of gas is allowed to escape? How about the partial pressure of O2? R= 0.08206\frac{L atm}{mol K} Homework Equations Using the ideal...

Homework Statement Homework EquationsThe Attempt at a Solution 1. If z=x+sin(##x^2##y) + ln y find ##\frac{\partial ^2z}{\partial x^2}## and ##\frac{\partial ^2z}{\partial y^2}## 2. Second order partial differentiation. 3. ##\frac {\partial z}{\partial x}## = 1 + ##cos(x^2y)## . (2x) =...

1) We know that when both liquid and vapor are present, and system of these is in phase equilibrium; the "partial pressure of the vapor" must be equal to the "vapor pressure" , i.e. : partial pressure of vapor= vapor pressure. 2) However, what happens if there is no liquid in the system, i.e...

(Wish there was a solutions manual...). Please check my workings below Show $ \int \frac{dz}{{z}^{2} + z} = 0 $ by separating integrand into partial fractions and applying Cauchy's Integral theorem for multiply connected regions. For 2 paths (i) |z| = R > 1 (ii) A square with corners $ \pm 2...

Textbook by Asmar. Would this class help me a lot for grad courses, like Jackson Electrodynamics or Sakurai Quantum? Debating to just finish up my upper levels and get As

Homework Statement Hi, I am reviewing a practice exam for my course and I am a bit stuck. "Assume that a sequence of partial sums (s_n) converges, can we also then say the sequence a_n is convergent? Does this statement go both ways? Answer: Yes, yes" The Attempt at a Solution On our exam...

Shouldn't the ΔG° partial pressure of the components be based on the K? Where ΔG°=-RTlnK such that K is based on the partial pressures of the gas involved? If we were to set the partial pressure to be 1bar each then every reaction having the same stoichiometric proportion of reactants and...

What is the general solution of the following hyperbolic partial differential equation: The head (h) at a specified distance (x) is a sort of a damping function in the form: Where, a, b, c and d are constants. And the derivatives are with respect to t (time) and x (distance). Thanks in advance.

Homework Statement Define f(x,y) = x+2y, w = x+y. What is ∂f / ∂w? Homework EquationsThe Attempt at a Solution f = w+y so: ∂f/∂w = ∂(w+y)/∂w = ∂w/∂w + ∂y/∂w = 1 + ∂y/∂w. But I'm really not sure if this is right and if it right so far, I can't figure out what ∂y/∂w should be...

If I have a function ##f(u,u^*) = \int u^* \hat{O} u d^3\mathbf{r}## both ##u## and ##u^*## are functions of ##\mathbf{r}## where ##\mathbf{r}## position vector, ##\hat{O}## some operation which involves ##\mathbf{r}## (e.g. differentiation), and the star sign denotes complex conjugate. Now I...

Homework Statement 60 L of N2 are collected over H2O at 40oC when atmospheric pressure is 760.00 torr. What is partial pressure of N2? Homework Equations PV=nRT Pt=P1+P2... Vapor pressure of H2O at 40oC:7.3590 KPa 760 torr=101.3 kPa 40oC=313oK The Attempt at a Solution PV-nRT...

$\int\frac{6{x}^{2}+22x-23} {(2x-1)(x+3)(x-2)} dx $ Solve using partial fractions $\frac{A}{2x-1}+\frac{B}{x+3}-\frac{C}{x-2}$ I pursued got A=2 B=-1 C=-3 Then?

Homework Statement Suppose we have an equation, ex + xy + x2 = 5 Find dy/dx Homework Equations Now I know all the linear differentiation stuff like product rule, chain rule etc. Also I know partial differentiation is differentiating one variable and keeping other one constant. The Attempt at...

Hey! :o Let $w=f(x, y)$ a two variable function and $x=u+v$, $y=u-v$. Show that $$\frac{\partial^2{w}}{\partial{u}\partial{v}}=\frac{\partial^2{w}}{\partial{x^2}}-\frac{\partial^2{w}}{\partial{y^2}}$$ I have done the following: We have $w(x(u,v), y(u, v))$. From the chain rule we have...

Homework Statement What is the partial fraction decomposition in ##\mathbb{R}[X]## of ##F = \frac{1}{X^{2n} - 1 } ##, ##n\ge 1##. Homework EquationsThe Attempt at a Solution Is this correct ? ## F = \frac{1}{2n}(\frac{1}{X-1} - \frac{1}{X+1} + 2 \sum_{k = 1}^{n-1} \frac{ \cos...

I would like to define t^*= \phi(r, t) given dt^* = \left( 1-\frac{k}{r} \right) dt + 0dr where k is a constant. Perhaps it doesn't exist. It appears so simple, yet I've been running around in circles. Any hints?

eifphysics submitted a new PF Insights post A Partial "Derivation" of Gauss's Law Continue reading the Original PF Insights Post.

Today, I had a class on Complex analysis and my professor wrote this on the board : The Laplacian satisfies this equation : where, So, how did he arrive at that equation?

Find the partial derivatives of the following function: Q=(1/3)logeL+(2/3)logeK Any help would be much appreciated! Below is my working out so far: \frac{\partial Q}{\partial L}= \frac{\frac{1}{3}}{L} \frac{\partial Q}{\partial K}= \frac{\frac{2}{3}}{L} Are these correct?

(x-3)/(x^2+4x+3) After i factor the denominator what do i do next to find A and B? =(x-3)/(x+3)(x+1) =A/(x+3)+B/(x+1)

Homework Statement I have two equations. cos(θ)wφ + sin(θ)wφ = 0 (1) And ## \frac{w_φ}{r}## + ∂wφ/∂r = 0 (2) Find wφ, which is a function of both r and theta. Homework EquationsThe Attempt at a Solution I end up with two equations, having integrated. wφ=## \frac{A}{sinθ}## from (1)...

Homework Statement There is a statement in a book : " Graph of P vs ##\chi## is a straight line which ##cannot## pass through origin" Homework EquationsThe Attempt at a Solution But if mole fraction of a component is zero then it can't form vapours because of which its partial pressure will be...

Can someone please explain to me why the expression ##[-\Box + U''(\Phi(r))]## is called the fluctuation operator? I was also wondering how to derive the following for the ##l^{th}## partial wave of the above operator: ##-\frac{d^2}{dr^2}-\frac{3}{r}\frac{d}{dr} + \frac{l(l+2)}{r}+...

Homework Statement Given: z = f(x,y) = x^2-y^2 To take the partial derivative of f with respect to x hold y constant then take the derivative of x. ∂f/∂x = 2x What I don't understand is why such would equal 2x, when the y is still there it just isn't variable and is ignored. Wouldn't it be...

My lecture notes give an example of two decay modes of ##K^+##, namely ##K^+\rightarrow \mu^+ \nu_\mu## and ##K^+\rightarrow e^+ \nu_e##. Both of these decays are suppressed due to helicity considerations which I understand, and the suppression factors are ##\frac{m_\mu c^2}{E_\mu}## and...

In the proof, mean value theorem is used (in the equal signs following A). Hence, the conditions for the theorem to be true would be as follows: 1. ##\varphi(y)## is continuous in the domain ##[b, b+h]## and differentiable in the domain ##(b, b+h),## and hence ##f(x,y)## is continuous in the...

In the solution to a differential-equation problem -- proof of the existence of an integrating factor -- the following statements are made regarding a general function "u(xy)" [that is, a function of two variable that depends exclusively on the single factor "x*y"]...

I am not seeing why my curve is not smooth. I normalized the data so it is not due to that. The partial LSE just assumes all other channels are part of the noise term (i.e. it performs worse than the full LSE model). clear all; close all; clc; %Parameters M = 5; %base station antennas K = 8...

I'm trying to prove that ##\sqrt{-g}\bigtriangledown_{\mu}v^{\mu}=\partial_{\mu}(\sqrt{-g}v^{\mu}) ## So i have ##\sqrt{-g}\bigtriangledown_{\mu}v^{\mu}=\sqrt{-g}(\partial_{\mu}v^{mu}+\Gamma^{\mu}_{\mu \alpha}v^{\alpha}) ## by just expanding out the definition of the covariant derivative...

Homework Statement Let f\colon\mathbb{R}^m\to\mathbb{R}. All partial derivatives of f are defined at point P_0\colon = (x_1, x_2, ... , x_m). If f has local extremum at P_0 prove that \frac{\partial f}{\partial x_j} (P_0) = 0, j\in \{1, 2, ..., m\} Homework Equations Fermat's theorem: Let...

Comparing two sources one has ##\frac{d^{2}x^{i}}{dt^{2}}=-\frac{1}{2}\epsilon\bigtriangledown_{i}h_{00} ## and the other has ##\frac{d^{2}d^{i}}{dt^{2}}=\frac{1}{2}\epsilon\bigtriangledown^{i}h^{00}##, And the one using the lower index has the Newton-Poisson equation as ##...

I am a little confused about; how to identify that Hadron/Meson may have S, P or D wave contribution to its decay to other hadrons. e.g in case of light meson decay a1→ ρπ this decay channel have two partial...

I am getting a little confused on which error propagation to use: I am looking to calculate the error in B*Cos(θ) , for the vertical axis of a williamson hall plot. where B is fwhm of a peak with it's own error and cos of the bragg angle I am unsure of whether i need to use partial derivative...

Question: The Breit-Wigner cross-section for a resonance R is ## \sigma_{i \to f} =12\pi\frac{\Gamma_{R\to i} \Gamma_{R\to f}}{(s-M^{2})^{2}+M^{2}_{R}\Gamma^{2}_{R total}}## [1], where ##s## is the com energy squared, ##M_{R}## is the mass of the resonance , ##\Gamma_{R total}## is the total...

Suppose we have a rational function ##P## defined by: $$P(x) = \frac{f(x)}{(x-a)(x-b)}$$ This is defined for all ##x##, except ##x = a## and ##x = b##. To decompose this function into partial fractions we do the following: $$\frac{f(x)}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}$$ Multiplying...

Homework Statement ∫ [x^(3)+4] / [x^(2)+4] dx Homework Equations N/A The Attempt at a Solution I know that the fraction is improper, so I used long division to rewrite it as x+(-4x+4)/[x^(2)+4]. Given the form S(x)+R(x)/Q(x), Q(x) is a distinct irreducible quadratic factor [x^(2)+4]. I used...

Homework Statement Given that the surface x^7y^2+y^4z^6+z^8x^8+9xyz=12 has the equation z=f(xy) in a neighbourhod of the point (1,1,1) with f(x,y) differentiable, find the derivatives. df/dx (1,1) = ? d^2f/dx^2 (1,1) = ? Homework EquationsThe Attempt at a Solution df/dx (1,1) I got -24/23 or...

Homework Statement a) Show that the function f(x,y)=\sqrt[3]{xy} is continuous and the partial derivatives f_x and f_y exist at the origin but the directional derivatives in all other directions do not exist b) Graph f near the origin and comment on how the graph confirms part (a). 2. The...

When can I do the following where ##h_{i}## is a function of ##(x_{1},...,x_{n})##? \frac{\partial}{\partial x_{k}}\frac{\partial f(h_{1},...,h_{n})}{\partial h_{m}}\overset{?}{=}\frac{\partial}{\partial h_{m}}\frac{\partial f(h_{1},...,h_{n})}{\partial x_{m}}\overset{\underbrace{chain\...

Homework Statement Find \frac{\partial}{\partial x} if: f(x,y) = \begin{cases}x^2\frac{\sin y}{y}, & y\neq 0\\0, &y=0 \end{cases} Homework EquationsThe Attempt at a Solution If y\neq 0 , then it's simple, but I get confused about the second part. How can I exactly utilize the limit definition...

Homework Statement Show that the coefficient of volume expansion can be expressed as β= -1÷ρ (∂ρ÷∂T) keeping P (pressure) constant Where rho is the density T is Temperature Homework Equations 1/v =ρ β= 1/v (∂v÷∂T) keeping P (pressure ) constant The Attempt at a Solution I started with...

1. Problem Define a function: for t>=0, f(x,t) = { x for 0 <= x <= sqrt(t), -x + 2sqrt(t) for sqrt(t) <= x <= 2sqrt(t), 0 elsewhere} for t<0 f(x,t) = - f(x,|t|) Show that f is continuous in R^2. Show that f_t (x, 0) = 0 for all x. Then define g(t) = integral[f(x,t)dx] from -1 to 1. Show...

Homework Statement Show that if f is homogeneous of degree n, then x\frac{\partial f}{\partial x} + y\frac{\partial f}{\partial y} = nf(x,y) Hint: use the Chain Rule to diff. f(tx,ty) wrt t. 2. The attempt at a solution I know that if f is homogeneous of degree n then t^nf(x,y) =...

If three variables x,y and z are related via some condition that can be expressed as $$F(x,y,z)=constant$$ then the partial derivatives of the functions are reciprocal, e.g. $$\frac{\partial x}{\partial y}=\frac{1}{\frac{\partial y}{\partial x}}$$ Is the correct way to prove this the following...

Homework Statement x(d^2y/dx^2)+dx/dt+xy=0 Homework EquationsThe Attempt at a Solution At first I thought it was an ODE, but then I found out the derivative was respect to to variables x and t. I am not sure if it is an ODE or PDE. What are the dependent and independent variables in the...