Partial Definition and 1000 Threads

Homework Statement Homework Equations General wave solution y=f(x+ct)+g(x-ct) [/B] The Attempt at a Solution [/B] Graphical sketch

Homework Statement If ## z=x^2+2y^2 ##, find the following partial derivative: \Big(\frac{∂z}{∂\theta}\Big)_x Homework Equations ## x=r cos(\theta), ~y=r sin(\theta),~r^2=x^2+y^2,~\theta=tan^{-1}\frac{y}{x} ## The Attempt at a Solution I've been using Boas for self-study and been working on...

Homework Statement The question asks to calculate ∂f/∂x for f(x,y,t) = 3x2 + 2xy + y1/2t -5xt where x(t) = t3 and y(t) = 2t5 Homework Equations The answer is given as ∂f/∂x = 6x + 2y - 5t The Attempt at a Solution I'm confused because the answer given seems to treat x,y ,t as...

If F = Fxi + Fyj +Fzk is a force field, do the following derivatives have physical significance and are they related to the components of the stress tensor? I notice they have the same dimensions as stress. ∂2Fx / ∂x2 ∂2Fx / ∂y2 ∂2Fx / ∂z2 ∂2Fx / ∂z ∂y ∂2Fx / ∂y ∂z ∂2Fx / ∂z ∂x ∂2Fx / ∂x...

Hi. If I have a function f ( x , t ) = x - 6t with x ( t ) = t2 and I take the partial derivative of f with respect to x I get the answer 1 as t acts as a constant so its derivative is zero. But if I substitute t with x1/2 I get the answer 1 - 3x-1/2 which is obviously different and wrong , I...

$\tiny{242 .10.09.8}\\$ $\textsf{Express the integrand as a sum of partial fractions and evaluate integral}$ \begin{align*}\displaystyle I&=\int f \, dx = \int\frac{\sqrt{16+5x}}{x} \, dx \end{align*} \begin{align*}\displaystyle f&=\frac{\sqrt{16+5x}}{x}...

Hi everyone, I am stuck on a problem. I need to give a partial fraction of 1/N(k-N). I have tried every method so far ( plotting roots, systems of equations). I think I found A=1/k but I have no clue how to find B value. I would really appreciate any help as I am a desperate student trying to...

Hello everybody. Consider $$\frac{\partial}{\partial t}f(x) + ax\frac{\partial }{\partial x}f(x) = b x^2\frac{\partial^2}{\partial x^2}f(x)$$ This is the equation (19) of...

Hi, I need to solve a system of first order partial differential equations with complex variables given by What software should I use for solving this problem..? The system includes 13 differential equations ...

$\tiny{206.07.05.88}$ \begin{align*} \displaystyle I_{88}&=\int\frac{1}{(x+2)\sqrt{x^2+4x+3}} \, dx \\ &=? \end{align*} would partial fractions be best for this?

What is the result of this kind of partial differentiation? \begin{equation*} \frac{\partial}{\partial x} \left(\frac{\partial x}{\partial t}\right) \end{equation*} Is it zero? Thank you in advance.

Its about functions with two or more variables ? How do you keep this x and y constant , i don't understand .

If the slope of the curve (derivative) at a given point is a number .

Hello all, If R is a partial order relation, is it true to say that \[R\cup R^{-1}\] \[R^{2}\] \[R\cap R^{-1}\] Are equivalence relations ? Regarding the first one, I think that the answer is yes. If \[xRx\] then it remains after the union. Asymmetry means that \[xRy\] without \[yRx\]...

Hello all, I have another question about partial order relations, again, a few statements which are either true or false. R is a partial order relation on a set A which is not necessarily finite. 1) With this order, A has at least one maximal and one minimal elements. 2) If with this order...

Hello, I have a question which includes several statements, which I need to decide if they are true or false. I am not sure how to do it, if you could give me hints or "leads", it will mostly appreciated. R is a partial order relation on A, a set of functions from [0,1] to [0,infinity) such...

I am having some trouble solving the problem shown below. Can anyone point me in the right direction? or provide the location of a worked example? The volume V of a cone of height h and base radius r is given by V=1/3 πr^2 h. The rate of change of its volume V due to stress expansions with...

If we consider function ##z=z(x,y)## then ##dz=(\frac{\partial z}{\partial x})_ydx+(\frac{\partial z}{\partial y})_xdy##. If ##z=const## then ##dz=0##. So, (\frac{\partial z}{\partial x})_ydx+(\frac{\partial z}{\partial y})_xdy=0 and from that \frac{dx}{dy}=-\frac{(\frac{\partial z}{\partial...

Homework Statement Show that ##\frac{\partial U}{\partial T}|_{N} = \frac{\partial U}{\partial T}|_{\mu} + \frac{\partial U}{\partial \mu}|_{T} \frac{\partial \mu}{\partial T}|_{N} ## (Pathria, 3rd Edition, pg. 197) Homework Equations ##U=TS + \mu N - pV## The Attempt at a Solution I tried to...

Hi all distinguished members of this forum ! As my first post here I would like to open with a situation I have been struggling with I have attached a simple diagram of my problem I have 2 cylindrial N52 neodymium magnets axially aligned with each other with opposite poles facing each other...

In Delayed choice quantum eraser experiment (https://en.wikipedia.org/wiki/Delayed_choice_quantum_eraser) entangled photons are sent on different paths. They reach their detectors at different times. The one arriving early is called a signal photon. The photon that arrives at its destination at...

I was reading a research paper, and I got stuck at this partial differentiation. Please check the image which I have uploaded. Now, I got stuck at Equation (13). How partial derivative was done, where does summation gone? Is it ok to do derivative wrt Pi where summation also includes Pi...

Homework Statement See below Homework EquationsThe Attempt at a Solution I am looking at a particular integral, and to get started, my text gives the indication that one should use partial fraction decomposition with ##\displaystyle \frac{\cos (ax)}{b^2 - x^2}##. Specifically, it says "then...

Homework Statement Find the partial fraction decomposition of ##\displaystyle \frac{1}{x^4 + 2x^2 \cosh (2 \alpha) + 1}## Homework EquationsThe Attempt at a Solution Using the identity ##\displaystyle \cosh (2 \alpha) = \frac{e^{2 \alpha} + e^{- 2\alpha}}{2}##, we can get the fraction to the...

Trouble here in the below partial fraction (Bug) $\frac{5x^2+1}{(3x+2)(x^2+3)}$ One factor in the denominator is a quadratic expression Split this into two parts A&B $\frac{5x^2+1}{(3x+2)(x^2+3)}=\frac{A}{(3x+2)}+\frac{Bx+c}{(x^2+3)}$...

Hello Everyone , I need some help in solving this partial fraction $\frac{x^2}{(x-2)(x+3)(x-1)}$ I am using this method in which the partial fraction is broken into 3 parts namely A,B &C $\frac{x^2}{(x-2)(x+3)(x-1)}=\frac{A}{(x-2)}+\frac{B}{(x+3)}+\frac{C}{(x-1)}$...

partial fractions $$\int\frac{3x^2+x+12}{(x^2+5)(x-3)} =\frac{A}{(x^2+5)}+\frac{B}{(x-3)}$$ $$3x^2+x+12=A(x-3)+B(x^2+5)$$ x=3 then 27+3+12=14B 3=B x=0 then 12=-3A+15 1=A $$\int\frac{1}{(x^2+5)} \, dx +3\int\frac{1}{(x-3)}\, dx$$ $\displaystyle...

Homework Statement Hi guys, I am having a problem, knowing where to start with this question. Before I spend trying derive the partial derivative chain rule from first principles I would just like to know if this is what this questions is asking. If it is not asking that, how do I go about...

Homework Statement In a sealed container is Helium ##M_{He} = \frac {4kg} {kmol} ## with a pressure of ## p_{He} = 4bar##. now is Methan put isothermic inside the container till both the methan and the helium mass are equal( ##M_{CH4} = \frac {16kg} {kmol} ## Calculate using the ideal gas law...

Homework Statement (a) Light waves satisfy the wave equation ##u_{tt}-c^2u_{xx}## where ##c## is the speed of light. Consider change of coordinates $$x'=x-Vt$$ $$t'=t$$ where V is a constant. Use the chain rule to show that ##u_x=u_{x'}## and ##u_{tt}=-Vu_{x'}+u_{t'}## Find ##u_{xx},u_{tt},##...

Hi PF! Regarding derivatives, suppose we have some function ##f = y(t)x +x^2## where ##y## is an implicit function of ##t## and ##x## is independent of ##t##. Isn't the following true, regarding the difference between a partial and full derivative? $$ \frac{df}{dt} = \frac{\partial f}{\partial...

Homework Statement Hi guys, I am having real trouble with the function 10ii) I can take the derivatives, but I feel like I am missing something, with what I have done. I set $f_x=0$and $f_y=0$ but really can't seem to find away to solve, i keep getting (0,0) which when I plug into wolfram it...

I have figured out a nice way to prove that if the complex numbers z_1,z_2,\ldots, z_N\in\mathbb{C} are all distinct, then the equation \prod_{n=1}^N \frac{1}{z - z_n} = \sum_{n=1}^N \frac{\alpha_n}{z-z_n} is true for all z\in\mathbb{C}\setminus\{z_1,z_2,\ldots, z_N\}, where the alpha...

Homework Statement Hi guys, I am have a problem with the question displayed below: [/B] Its 6.1 ii) I am really not sure how I am suppose to approach this. I am new to partials, so any advice would be great. Homework EquationsThe Attempt at a Solution So far I have: $$\frac{\partial ^2...

$\tiny{242t.8.5.9}$ $\textsf{expand the quotient by}$ $\textbf{ partial fractions}$ \begin{align*}\displaystyle y&=\int\frac{dx}{9-25x^2} &\tiny{(1)}\\ \end{align*} $\textit{expand and multiply every term by $(3+5x)(3-5x)$}$ \begin{align*}\displaystyle...

Homework Statement I am trying to wrap my head around what it means to find an explicit formula for the sequence of partial sums. Question: Find an explicit formula for the sequence of partial sums and determine if the series converges. a) sum from n=1 to n=infinity of 1/(n(n+1)) Homework...

Let's say I have two vector fields a(x,y,z) and b(x,y,z). Let's say I have a scalar field f equal to a•b. I want to find a clean-looking, simple way to express the directional derivative of this dot product along a, considering only changes in b. Ideally, I would like to be able to express...

Homework Statement Can someone please check my working, as I am new to Einstein notation: Calculate $$\partial^\mu x^2.$$ Homework Equations 3. The Attempt at a Solution [/B] \begin{align*} \partial^\mu x^2 &= \partial^\mu(x_\nu x^\nu) \\ &= x^a\partial^\mu x_a + x_b\partial^\mu x^b \ \...

My physics book is showing an example of why it matters "what variable you hold fixed" when taking the partial derivative. So it asks to show that ##(\frac{\partial{w}}{\partial{x}})_{y} \neq (\frac{\partial{w}}{\partial{x}})_z## where ##w=xy## and ##x=yz## and the subscripts are what variable...

$\tiny{206.08.05.44}$ $\textsf{Use the method of partial fraction decomposition}\\$ \begin{align*} \displaystyle I_{44} &=\int \frac{4x^3+6x^2+128x}{x^5+32x^3+256x}dx\\ &=4\int \frac{1}{x^2+16} \, dx +6\int \frac{x}{(x^2+16)^2} \, dx +64\int \frac{1}{(x^2+16)^2} \, dx \end{align*}$\textsf{so far...

Homework Statement The equation is z= e (x*y), the interesting thing is y is function of x too, y = ψ(x) Calculate the partial derivative respect to x, and the total derivative. Homework Equations Total differential: dz= ∂z/∂x dx + ∂z/∂y dy The Attempt at a Solution [/B] Well, according...

Hi there. I am trying to self teach how to solve partial differential equations numerically using finite differences. I know this is a complex field, that requires much more knowledge of the theory than what I actually know, but anyway I wanted to try. Anyway, I've tried to build my own...

Hello everyone. So I wanted to get some opinions on what some of you thought was a better choice, as far taking PDE's or classical mechanics 2 goes. First let me start off by giving a little info; I've already taken calc 1-3 and ordinary differential equations, physics 1 & 2...

Homework Statement Consider the Laplace Equation of a semi-infinite strip such that 0<x< π and y>0, with the following boundary conditions: \begin{equation} \frac{\partial u}{\partial x} (0, y) = \frac{\partial u}{\partial x} (0,\pi) = 0 \end{equation} \begin{equation} u(x,0) = cos(x)...

$\tiny{206.8.5,42}\\$ $\textsf{partial fraction decomostion}\\$ \begin{align} \displaystyle && I_{42}&=\int\frac{3x^2+x-18}{x^3+9x}\, dx& &(1)&\\ && \frac{3x^2+x-18}{x^3+9x} &=\frac{Ax+B}{x^2+9} +\frac{C}{x} & &(2)& \end{align} $\textit{just seeing if this is set up ok before finding values} $

Homework Statement and the solution (just to check my work) Homework Equations None specifically. There seems to be many ways to solve these problems, but the one used in class seemed to be partial fractions and Taylor series. The Attempt at a Solution The first step seems to be expanding...

$\tiny{206.8.5.49}$ $\textsf{Express the integrand as sum of partial fractions}$ \begin{align} && I_{49}&=\int\frac{30s+30}{(s^2+1)(s-1)^3}\, ds& &(1)& \\ &\textsf{expand}& \\ && &=\displaystyle 15\int\frac{1}{(s^2+1)}\, ds -15\int\frac{1}{(s-1)^2}\, ds +30\int\frac{1}{(s-1)^3}\, ds&...

Is there a good description or formula regarding how the sound pressure from a constant source depends upon ambient pressure? That is, if I were to conduct an experiment where I put a source and a microphone in a container, and then change the pressure in that container with a pump, assuming...

Homework Statement Homework Equations The Attempt at a Solution Because we are only looking at a cross section, I tried to reduce 5.3 down to just being a function of R and Theta. However I reasoned that there should be, based on this problem, no dependence on Theta either, so I figured I...

Homework Statement So I know I have to take the derivative with respect to x, then respect to y, then respect to z, but I am not getting the right answer. I know that the answer is 0 and my professor did it with very few steps that I do not understand. Can someone please guide me through it?