Partial Definition and 1000 Threads

Homework Statement f(x) = f(x_1,...,x_N) : R^N \mapsto R has continuous partial derivatives. Assume that for a point a in R^N , \frac{\partial f}{\partial x_i}(a) \neq 0 for all i = 1...N. The implicit function theorem says that near a, the equation f(x) = f(a) can be used to express each...

Homework Statement S is a ball of radius 1 in R^2; Δu=0 in S u=g in ∂S, g(x1,x2)>1 for any (x1,x2) in ∂S. Show that for any r satisfying 0<r<1 there is a point (x1,x2) in S such that u(x1, x2) >=1. Homework Equations using mean value formula: ∫u(y)dy=1/Vr^n(∫u(y)dy) The...

Hi all! I have the following slide, and whilst I understand that the original point is "the rate of density, ρ, in each volume element is equal to the mass flux"...i am totally lost on the mathematics! (And I am meant to be teaching this tomorrow). I do not have any information on what the...

Hi everyone, We just started learning partial derivatives and I understand the fx notation, but I'm confused when I'm asked for the value of fxy. Does this mean multiply the two derivatives together? For example: What is fxy when f(x,y) = (x+2y)ln(xy) Thanks!

Homework Statement Evaluate the integral. Homework Equations \displaystyle\int_0^∞ {\frac{dv}{(1+v^2)(1+arctanv)}} The Attempt at a Solution i am not sure how to do partial fractions with an inverse trig function in the denominator. i tried v=arctanv so i could do a substitution...

Homework Statement Given: https://www.physicsforums.com/attachments/56653, show that this can be written as: https://www.physicsforums.com/attachments/56651. Homework Equations Hint: https://www.physicsforums.com/attachments/56652 The Attempt at a Solution Quite confused by this...

Homework Statement Find df/dx, f(x,y)=integral of sqrt(1-t^3)dt from x^2 to x^3. Since it is asking to find the derivative with respect to x,should I regard t as a constant? Homework Equations The Attempt at a Solution I tried to find the antiderivative of the integral...

Hi all, I'm trying to figure out the following problem: Find df/dx, f(x,y)=integral of sqrt(1-t^3)dt from x^2 to x^3. Since it is asking to find the derivative with respect to x,should I regard t as a constant?

Hello PH, This is my first post. I came here while studying partial derivatives and after clicking here and there for over 4hrs for an answer. While practicing the derivatives rules i came across the hideous quotient rule. I've solved around 20 fractional problems trying to find a decision...

Homework Statement This is an arc length problem in three dimensions. I was given the vector r(t)=<et, 1, t> from t=0 to t=1 Homework Equations Arc Length= \int |\sqrt{r'(t)}| dt from t1 to t2 where |\sqrt{r'(t)}| is the magnitude of the derivative of the vector The Attempt at a...

The idea of varying one thing but keeping others constant is central in scientific analysis. People want to know, other things constant, the effect of taking vitamins, smoking or drinking alcohol, just as examples. Is the idea of the partial derivative analogous to scientific empiricism's...

{\frac{∂(xy)}{∂x}=x} Going backwards. If we took, ∫x dy we get xy+f(x) Now, the only way that ∫x dy is a valid operation, is if we know that we came from a partial derivative. Why, when taking a partial...

1. ∫[(x4 + x + 1)/(x(x2 + 1))]dx 2. When I first did this problem, I divided and got: ∫[x + (-x2 + x + 1)/(x3 + x)]dx (x3 + x) = x(x2 + 1) I then set up the fraction as: A/x + B/(x2 + 1) BUT, the solution to this problem says: A/x + [(Bx + C)/(x2 + 1)] How would I know to use...

Here is the question: Here is a link to the question: Decompose the equation into two simpler fractions? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Homework Statement Integrate (3x^2-10)/(x^2-4x+4) dx using partial fractions. Homework Equations None The Attempt at a Solution I tried using A/(x-2) + B/(x-2)^2 but I didnt get a coeffecient of an x^2. I've also tried using (Ax+B)/(x-2) + C/(x-2)^2 Though I...

Homework Statement when steam was heated with excess carbon in a closed container to 1800kpa, the partial pressure of steam at equilibrium of steam was 318kpa, find Kp H2 ,Kp CO and find Kp Homework Equations H2O(g) + C(s) -> H2(g) + CO(g) <-...

Homework Statement Find a value for n for which the nth partial sum is ensured to approximate the sum of the alternating harmonic infinite series to three decimal places. Homework Equations Sn = Ʃ(-1)^k+1*1/k = 1 - 1/2 + 1/3 - 1/4 + 1/5 - . . . S1 = 1 S2 = 1 - 1/2 S3 = 1 - 1/2 + 1/3 S4...

Homework Statement For the heat equation u_{t}=\alpha^{2}u_{xx} for x\in\mathbb{R} and t>0, show that if u(x,t) is a strong solution to the heat equation, then u_{t} and u_{x} are also solutions. Homework Equations u_{t}=\alpha^{2}u_{xx} The Attempt at a Solution I've considered...

Homework Statement general course question Homework Equations N/A The Attempt at a Solution fx is a first order partial derivative fxy is a second order partial derivative fxyz is a third order partial derivative I understand that Clairaut's Theorem applies to second order...

Homework Statement Given that z = √3x/y show that: Homework Equations ∂2z/∂x∂y = ∂2z/∂y∂x The Attempt at a Solution

Homework Statement Given the function f(x,y)=\frac{1}{2x^2 + y} Find the partial derivative fxx(x,y) Homework Equations The Attempt at a Solution Seems pretty straight forward, just treat y as a constant and differentiate twice. But I keep getting the answer wrong and I have...

When discussing the second partial derivative test in multivariate calculus, a reference is usually made to an elusive "higher order test" that one must defer to in the case that the second partial derivative test fails. Does anyone know the general form of these higher order test? My first...

How is the double derivative equal to that in the equation 2 in the attachment? =|

I am utilitizing rotation vectors (or SORA rotations if you care to call them that) as a means of splitting 3D rotations into three scalar orthogonal variables which are impervious to gimbal lock. (see SO(3)) These variables are exposed to a least-squares optimization algorithm which...

Here is the question: Here is a link to the question: Help with Calculus BC: partial fractions!? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

In a thermodynamics question, I was recently perplexed slightly by some partial derivative questions, both on notation and on physical meaning. I believe my questions are best posed as examples. Suppose we have an equation, (\frac{\partial x(t)}{\partial t}) = \frac{1}{y}, where y is a...

Homework Statement Suppose c_{n} > 0 for each n\geq 0. Prove that if \sum ^{\infty}_{n=0} c_{n} is Cesaro summable, then the partial sums S_{N} are bounded. Homework Equations -- The Attempt at a Solution I tried contraposition; that was getting me nowhere. I have a few...

1. So, i have the next integrand... 2. \int \frac{1}{(x-1)^2(x+1)^2}\,dx 3. I proceeded by resolving it by partial fraction and i came up with the next... \int \frac{1}{((x-1)^2)((x+1)^2)}\,dx = \int \frac{A}{(x-1)} + \frac{B}{(x-1)^2} + \frac{C}{(x+1)} + \frac{D}{(x+1)^2}\,dx The thing is...

Homework Statement Use the method of partial fractions to show that: $$\frac{2x^2}{(1-x(1+x)} $$ , may be written as: $$-2+\frac{1}{1-x}+\frac{1}{1+x}$$ , where $$\lvert x\rvert\neq1 $$. Homework Equations The Attempt at a Solution I obviously know how to do it but in the...

Author: David Bleecker (Author), George Csordas (Author) Title: Basic Partial Differential Equations Amazon Link: https://www.amazon.com/dp/1571460365/?tag=pfamazon01-20 Prerequisities: Table of Contents: Preface Review and Introduction A Review of Ordinary Differential Equations...

Homework Statement compute the gradient: ln(z / (sqrt(x^2-y^2)) Homework Equations ∇=(∂/(∂x)) + ... for y and z I just want to know how to do the first term with respect to x The Attempt at a Solution I am so rusty I don't know where to begin.

Homework Statement The function f(x,y,z) may be expressed in new coordinates as g(u,v,w). Prove this general result: The Attempt at a Solution df = (∂f/∂x)dx + (∂f/∂y)dy + (∂f/∂z)dz dg = (∂g/∂u)du + (∂g/∂v)dv + (∂g/∂w)dw df = dg since they are the same thing? but the...

I've been trying to get out this question for a while now: ai) Show that (x,y,z) = (1,1,1) is a solution to the following system of equations: x + y + z = 3 2x + 2y + 2z = 6 3x + 3y +3z = 9 aii) Hence find the general solution of the system b) Express 2x^2 + 3/(x^2 + 1)^2 in partial...

Plese give me silminer simple example or anther example on this case or explein the steps

I have the following equation \frac{\partial}{\partial y}\left(m\frac{dy}{dx}\right)=0 where y is a function of x and m is a function of y. If I integrate this equation first with respect to y should I get a function of x as the constant of integration (say C\left(x\right)) or it is just...

Homework Statement f\in L_{loc}^1(\mathbb{R}_+). Need show that for Re(z)>\sigma_f (abscissa of absolute convergence) we have $$\mathcal{L}[tf(t)](z)=-\frac{d}{dz}\mathcal{L}(z)$$where \mathcal{L} denotes Laplace transform. The Attempt at a Solution The proof comes down to whether...

let f = x2 + 2y2 and x = rcos(\theta), y = rsin(\theta) . i have \frac{\partial f}{\partial y} (while holding x constant) = 4y . and \frac{\partial f}{\partial y} (while holding r constant) = 2y . i found these partial derivatives by expressing f in terms of only x and y, and then in...

Homework Statement x3 + y3 + z3 - 3xyz = 6 Find (∂y/∂x)z. Homework Equations [b]3. The Attempt at a Solution [/ can i simply take the partial derivative of both sides treating z as constant? x3 + y3 + z3 - 3xyz - 6 = 0 f(x,y,z) = 0 (∂f/∂x)z = 0

Hi everyone, I am working on simplifying a differential equation, and I am trying to figure out if a simplification is valid. Specifically, I'm trying to determine if: \frac{\del^2 p(x)}{\del p(x) \del x} = \frac{\del^2 p(x)}{\del x \del p(x)} where p(x) is a function of x. Both p(x)...

Homework Statement Given that u(x,y) and y(x,z) are both continuous, differentiable functions show that (\frac{\partial u}{\partial z})x=(\frac{\partial u}{\partial y})x(\frac{\partial y}{\partial z})x Homework Equations Only equations given above The Attempt at a Solution I...

Homework Statement Determine whether a function with partial derivatives f_x(x,y)=x+4y and f_y(x+y)=3x-y exist. The Attempt at a Solution The method I've seen is to integrate f_x with respect to x, differentiate with respect to y, set it equal to the given f_y and show that it can't be...

I'm working on a calculus project and I can't seem to work through this next part... I need to substitute equation (2) into equation (1): (1): r\frac{\partial}{\partial r}(r\frac{\partial T}{\partial r})+\frac{\partial ^{2}T}{\partial\Theta^{2}}=0 (2): \frac{T-T_{0}}{T_{0}}=A_{0}+\sum from n=1...

I have an electrostatics problem (shown here: https://www.physicsforums.com/showthread.php?t=654877) which leads to the following system of differential equations: \frac{\partial E_z}{\partial z}=\frac{\rho}{\epsilon_0} (1) Z_i E_r \frac{\partial \rho}{\partial r}+(u_g+ Z_i E_z)...

Hi guys, attached is a picture of my problem and it is also underlined. I've been reading through this theory and I just don't understand what the square brackets indicate. I understand that ∇phi is the partial derivative with respect to the scalar function phi. But what is ∇phi...

Homework Statement Given the initial value problem: \frac{(u)}{(1-e^-(2x))}u_{x}+ \frac{\sqrt{t}}{u}u_{t}=1, with x, t, u > 0 Subject to condition u(x,1)=e^{-x} Homework Equations a) Classify given partial differential equation. b) Write the characteristic equations. By...

Homework Statement How to get partial fraction decomposition for \frac{1}{(x^2+a^2)(x^2+p^2)}Homework Equations The Attempt at a Solution I tried with \frac{1}{(x+ia)(x-ia)(x+ip)(x-ip)}=\frac{A}{x+ia}+\frac{B}{x-ia}+\frac{C}{x-ip}+\frac{D}{x+ip} and get the result at the end of the day. Is...

Homework Statement At 100 o C Kc=.078 for the reaction SO2Cl2<-->SO2 + Cl2. In an equilibrium mixture the [SO2CL2]=.0108 M and [SO2]=.052 M. What is the partial pressure of Cl2 in the eq. mixture? Homework Equations Kp=Kc(RT)\Deltan P=RT/V The Attempt at a Solution I solve for...

I have a question to ask, is dx = δx, can they cancel each other like \frac{dx}{δx}=1 and is it mean that: \frac{δf}{δx}\frac{dx}{dt}=\frac{df}{dt}? (f = f (x,y,z))

Homework Statement Consider the following equality: (\frac{∂S}{∂V})T = (\frac{∂P}{∂T})V If I rearrange the equality so that I write: (\frac{∂S}{∂P})? = (\frac{∂V}{∂T})? What variables will be constant in each side? I'm having some trouble in a few thermodynamics problems because...

Homework Statement Consider the following equality: (\frac{∂S}{∂V})T = (\frac{∂P}{∂T})V If I rearrange the equality so that I write: (\frac{∂S}{∂P})? = (\frac{∂V}{∂T})? What variables will be constant in each side? I'm having some trouble in a few thermodynamics problems because...