Partial Definition and 1000 Threads

Find surface of $\begin{array}{l} \text{Problem Cauchy} \\ {a^2} \cdot {x_2} \cdot u \cdot {u_{{x_1}}} + {b^2} \cdot {x_1} \cdot u \cdot {u_{{x_2}}} = 2{c^2}{x_1}{x_2}{\rm{ }} \\ \end{array}$ The partial differntial equation passes through ${\rm{ C: = \{ }}\frac{{{x^2}}}{{{a^2}}} +...

What is the difference between partial derivatives and gradients, if there is any? I'm asking because I need to derive a function " f (T,P) " for air convection; where T is temperature and P is pressure and both are variables in this case. Thanks

Homework Statement Let l, w, and h be the length, width and height of a rectangular box. The length l is increasing with time at at rate of 1 m/s, while the width and the height are decreasing at rates 2 m/s and 1m/s respectively. At a certain moment in time the dimensions of the box are l=5...

I have a multivariable function z = x2 + 2y2 such that x = rcos(t) and y = rsin(t). I was asked to find (I know the d's should technically be curly, but I am not the best at LaTeX). I thought this would just be a simple application of chain rule: ∂2/(∂y∂t) = (∂z/∂x)(ⅆx/ⅆt) + (∂z/∂y)(ⅆy/ⅆt)...

Homework Statement Find the Inverse laplace transform of: http://www4c.wolframalpha.com/Calculate/MSP/MSP14541hg721e74730d4fb00004644i96f59549h1d?MSPStoreType=image/gif&s=30&w=201.&h=40. Result...

From the resonance structures of Naphthalene, 1-2 Bond has more double Bond character than 2-3 Bond. In March's Advanced organic chemistry, its given that ozone preferentially reacts with 1-2 Bond. But the reaction is not given. Is this a normal ozonolysis reaction in which the 1-2 Bond is...

Homework Statement What are the partial half of 22Na for decay by a)Ec b) β+ emission Homework Equations λ=ln2/T1/2 The Attempt at a Solution this what I do T1/2 =2.602 Yr λ=ln2/2.602 λ=0.266 yr-1what is the difference between a)Ec b) β+ emission there is no Percentage of each decay type.!

Hi, I'm a little confused about something. I have an object, and I want to take the partial derivative of its position wrt velocity and vice versa. I'm not sure how to begin solving this problem. Essentially, what I have is this: ## \frac{\partial x}{\partial \dot x} ## and ## \frac{\partial...

Homework Statement Find the equation of state given that k = aT^(3) / P^2 (compressibility) and B = bT^(2) / P (expansivity) and the ratio, a/b? Homework Equations B = 1/v (DV /DT)Pressure constant ; k = -1/v (DV /DP)Temperature constant D= Partial derivative dV = BVdT -kVdP (1) ANSWER is...

Homework Statement Homework Equations trigonometric identities The Attempt at a Solution I did a trig substitution of u=tan(θ/2) and from that I could substitute cos(θ) = 1-u2/1+u2 dθ = 2/(1+u2) du = 1/2 sec2(θ/2) dθ I simplified a bit and changed the bounds to get 2du/(5u2 + 1)(1 + u2)2...

Hey all, I am reading Goldstein and I am at a point where I can't follow along. He has started with D'Alembert's Principle and he is showing that Lagrange's equation can be derived from it. He states the chain rule for partial differentiation: \frac{d\textbf{r}_i}{dt}=\sum_k \frac{\partial...

Mod note: Moved from a homework section 1. Homework Statement N/A Homework Equations f(x + Δx,y) = f(x,y) + ∂f(x,y)/∂x*Δx The Attempt at a Solution Sorry this isn't really homework. We were given this equation today in order to derive the Taylor expansion formula in two variables and I'm not...

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...

Homework Statement A beam of neutrons hit a target of heavy nuclei with spin ##J_N = 0## with resonance when the energy of the incident beam is 250eV, in the cross section distribution with a maximum of 1300 barns. The width of the maximum is 20 eV. Find the partial width of resonance for the...

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...

Homework Statement is this statement is true : ##\nabla_\mu \nabla_\nu \sqrt{g} \phi = \partial_\mu \sqrt{g} \partial_\nu \phi## Homework EquationsThe Attempt at a Solution well we know ##\nabla_\mu \sqrt{g} =0## so it moves back : ## \nabla_\mu \sqrt{g} \nabla_\nu \phi =\sqrt{g} \nabla_\mu...

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 ##...