Partial Definition and 1000 Threads

My attempt I calculated the partial derivatives of n wrt P and T. They are given below. ##\frac {\partial n}{\partial P} = \frac{nb -1}{\left(2an-Pb-3abn^2-kT\right )}## ##\frac {\partial n}{\partial T}= \frac {nk}{\left(2an-Pb-3abn^2-kT \right ) }## I know that if the partial derivative is...

Let us suppose we have a functional of f such that ##f=f((\vec{r}(t),t)## where ##\vec{r}(t) = a(t)\vec{x}(t)##. I am trying to derive an equation such that $$\left.\frac{\partial}{\partial t}\right|_r = \left.\frac{\partial }{\partial t}\right|_x + \left.\frac{\partial \vec{x}}{\partial...

Hi, I just need some (hopefully) quick calculus help. I have the following: ##(y\frac {\partial } {\partial z}(z\frac{\partial f} {\partial x}))## After it is expanded this is the solution: ##(yz\frac {\partial^2 f} {\partial z \partial x} + y\frac{\partial f} {\partial x} \frac{\partial z}...

Moved from technical math section, so missing the homework template Summary:: Find a general formula for the nth derivative Hi everyone! How would I approach and answer a Q such as this I began by rewriting the expression in a different form, then used chain rule to each given term I...

The function should use (r,z,t) variables The domain is (0,H) Since U is not dependent on angle, then theta can be ignored in the expression for Laplacian in cylindrical coordinates(?)

In this example, DiscretePlot[ Sum[ f[x], {x,1,n} ],{n,1,20}] will Mathematica automatically optimize the procedure -- i.e., will it run a single loop where it calculates the sum up to 20 only once, transferring the partial sums to the output as it goes along? Assume that there is no...

I am attempting to understand how POVMs fit in with quantum measurement, and I think I am getting tripped up in notation when it comes to multipartite systems. The situation is as follows: System: \rho_A Measurement instrument: \rho_B = |\phi\rangle\langle\phi| (pure state) The multipartite...

Solve the boundary value problem Given u_{t}=u_{xx} u(0, t) = u(\pi ,t)=0 u(x, 0) = f(x) f(x)=\left\{\begin{matrix} x; 0 < x < \frac{\pi}{2}\\ \pi-x; \frac{\pi}{2} < x < \pi \end{matrix}\right. L is π - 0=π λ = α2 since 0 and -α lead to trivial solutions Let u = XT X{T}'={X}''T...

Do anyone know how to find ##1##, ##2x - 5##, and ##2\sqrt{x^2 - 5x + 6}## in the triangle? (please see attached image) Also, how do you find ##(x - 5/2)^2 - (1/2)^2##? [Moderator's note: Moved from a technical forum and thus no template.]

In the chemical engineering text of Smith, VanNess, and Abbott, there is a section on partial molar volume. It states that Gibbs theorem applies to any partial molar property with the exception of volume. Why is volume different? In other words, when evaluating the partial molar volume of a...

Consider the following linear first-order PDE, Find the solution φ(x,y) by choosing a suitable boundary condition for the case f(x,y)=y and g(x,y)=x. --------------------------------------------------------------------------- The equation above is the PDE I have to solve and I denoted the...

Introducing the new variables ##u## and ##v##, the chain rule gives ##\dfrac{{\partial{f}}}{{\partial{x}}}=\dfrac{{\partial{f}}}{{\partial{u}}} \dfrac{{\partial{u}}}{{\partial{x}}}+\dfrac{{\partial{f}}}{{\partial{v}}} \dfrac{{\partial{v}}}{{\partial{x}}}##...

If this belongs in classical physics, please move it there. But it seems like the kind of question chemistry people would know so I'm putting it here. I was reading a textbook on chemical thermodynamics, and it says to raise the partial molar Gibbs free energy of n moles a substance from...

Is there such a thing as a partial temperature of a gas in a mixture? Partial pressure is commonly accounted for and used. It seems that if there are molecules of different masses colliding in a mixture, their average respective velocities in a mixture should be different based on transfer of...

I apologise for the length of this question. It is probably possible to answer it by reading the first few lines. I fear I have made a childish error: I am working on the geodesic equation for the surface of a sphere. While doing so I come across the partial derivative \begin{align}...

I have this function, and I want to take the derivative. It includes a unit step function where the input changes with time. I am having a hard time taking the derivative because the derivative of the unit step is infinity. Can anyone help me? ##S(t) = \sum_{j=1}^N I(R_j(t)) a_j\\ I(R_j) =...

While reproducing a research paper, I came across the following equation, ∂f/∂t−(H(f)(∂f/∂x)=0 where [H(f)] is hilbert transform of 'f.' and f=f(x,t) and initial condition is f(x,0)=cos(x) and also has periodic boundary conditions given by F{H{f(x′,t)}}=i⋅sgn(k)F{f(x,t)}, where F(f(x,t) is...

If given a function ##u(x,y) v(x,y)## then is it correct to write ##\frac{\partial }{\partial x}u(x,y)v(x,y)=\frac{u(x+dx,y)v(x+dx,y)-u(x,y)v(x,y)}{dx}##??

Problem Statement: Use the definition of the total time derivative to a) show that ##(∂ /∂q)(d/dt)f(q,q˙,t) = (d /dt)(∂/∂q)f(q,q˙,t)## i.e. these derivatives commute for any function ##f = f(q, q˙,t)##. Relevant Equations: My approach is given below. Please tell if it is correct and if not ...

I was doing this problem from Griffith's electrodynamics book and can't figure out how to do this integral. The author suggested partial fractions but the denominator has a fractional exponent which I have never seen for partial fractions, and so, I am unsure how to proceed. The integral I am...

I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume I: Foundations and Elementary Real Analysis ... ... I am focused on Chapter 1: The Axioms of Set Theory ... ... I need some help to clarify an aspect of Garling's definition of a partial order...

It seems that the way to combine the information given is z = f ( g ( (3r^3 - s^2), (re^s) ) ) we know that the multi-variable chain rule is (dz/dr) = (dz/dx)* dx/dr + (dz/dy)*dy/dr and (dz/ds) = (dz/dx)* dx/ds + (dz/dy)*dy/ds ---(Parentheses indicate partial derivative) other perhaps...

I started by substituting the following anzatz: $$ \psi = \psi_e + \psi_1 $$ When ## |\psi_1| \ll 1 ##. Substituting the above into the equation yields: $$ \frac {d\psi_1} {dt} = -(1 + i\alpha)\psi_1 + \frac i 2 \frac {\partial ^ 2 \psi _1 } {\partial x ^ 2 } + i (\bar \psi_1 \psi_1 ^2 + \bar...

How do I interpret geometrically the partial derivative in respect to a constant of a function such as ##\frac{ \partial}{\partial c} (acos(x) + be^x + c)^2##?

I am given the following: $$u = (x,t)$$ $$\frac{\partial^2 u}{\partial t^2} - c^2\frac{\partial^2 u}{\partial x^2} = 0$$ and $$E = x + ct$$ $$n = x - ct$$ I need to solve for $$\frac{\partial^2 u}{\partial x^2}$$ and $$\frac{\partial^2 u}{\partial t^2}$$ using the chain rule.How would I even...

I have been reading a book on classical theoretical physics and it claims: -------------- If a Lagrange function depends on a continuous parameter ##\lambda##, then also the generalized momentum ##p_i = \frac{\partial L}{\partial\dot{q}_i}## depends on ##\lambda##, also the velocity...

Hello, I have to solve this second order differential equation. It's like a string vibrating equation but with a constant c: $$\frac{{\partial^2 u}}{{\partial t^2}}=k\frac{{\partial^2 u}}{{\partial x^2}}+c$$ B.C $$u(0,t)=0$$ $$u(1,t)=2c_0$$ c_0 is also a constant I.C $$u(x,0)=c_0(1-\cos\pi...

Hi all, I have had the following question in my head for quite a while: Thermodynamic potentials written in differential form look like $$dU = TdS - PdV$$ and we can obtain equations for say, temperature by doing the following partial $$T = \frac {\partial U}{\partial S} |_V$$ Does this mean...

Help please, I need to solve this differential equation x\frac{\partial^2 U}{\partial x^2}+y\frac{\partial^2 U}{\partial y^2}=aU in Matlab (where "a" is a constant parameter, it can be taken by any), I wanted to use the Partial Differential Equation Toolbox, but I ran into a problem, the...

When using the separation of variable for partial differential equations, we assume the solution takes the form u(x,t) = v(x)*g(t). What is the justification for this?

At the end of a long proof I came across something in tensor calculus that seems too good to be true. And if something seems too good to be true ... The something is that a second order partial derivative vanishes if one of the parts in the denominator is in the same reference frame as the...

Homework Statement Kc = 4.15 x 10-2 at 356°C for PCl5(g) ↔ PCl3(g) + Cl2(g). A closed 2.00 L vessel initially contians 0.100 mol PCl5. Calculate the total pressure in the vessel (in atm to 2 decimal places) at 356°C when equilibrium is achieved. Homework Equations PV=nRT Kp= Kc(RT)^change in...

Homework Statement I'm given a gas equation, ##PV = -RT e^{x/VRT}##, where ##x## and ##R## are constants. I'm told to find ##\Big(\frac{\partial P}{\partial V}\Big)_T##. I'm not sure what that subscript ##T## means? Homework Equations ##PV = -RT e^{x/VRT}## Thanks a lot in advance.

<Moderator's note: Moved from a technical forum and thus no template.> $$\lim_{x \to 0} \cos(\pi/2\cos(x))/x^2$$ I tried to evaluate the limit this way, $$\lim_{x \to 0} \cos(\pi/2\cdot1)/x^2$$ since $$\cos0=1$$ $$\lim_{x \to 0} \cos(\pi/2\cdot1)/x^2=\lim_{x \to 0} 0/x^2$$ Now apply...

Homework Statement This question relates to a very large project I have been assigned in a course on mathematical methods in structural engineering. I have to solve the following equation, in a specific way: (17) Now we have to assume the following solution: (18) It wants me insert...

I've a system of partial diff. eqs. in thermo-elasticity, I can solve it using normal mode analysis method but I need to solve it using laplace or Fourier

Suppose we have to deal with the question : $$\frac{\partial}{\partial x}\frac{\partial}{\partial y}=?\frac{\partial}{\partial y}\frac{\partial}{\partial x}$$ This seems true for independent variables. But if at the end x and y are linked in some way like $$x=f(t),y=g(t)$$ this is no more the...

Salutations, I have been trying to approach a modelling case about organism propagation which reproducing with velocity $$\alpha$$ spreading randomly according these equations: $$\frac{du(x,t)}{dt}=k\frac{d^2u}{dx^2} +\alpha u(x,t)\\\ \\ u(x,0)=\delta(x)\\\ \lim\limits_{x \to \pm\infty}...

Hello. Glad to meet you, everyone I am studying the [Mathematical Methods for Physicists; A Comprehensive Guide (7th ed.) - George B. Arfken, Hans J. Weber, Frank E. Harris] In Divergence of Vector Field, I do not understand that How to transform the equation in left side into that in right...

I am dealing with an expression in a large amount of literature usually presented as: \frac{\partial}{\partial \phi_i}\left(\nabla \phi_i \cdot \nabla \phi_j \right) I'm looking at tables of vector calculus identities and cannot seem to find one for the exact expression given, even if I...

Hi, I would like to expand the following expression: 1/[((a+s)*(1+b/s)^m)], where a, b, and s are real nonnegative values and m is an arbitrary positive integer. Particularly, according to partial fraction expansion, it becomes: Sum[A_j/[(1+b/s)^j],{j,1,m}]+B/(a+s). I look for a closed-form...

4\frac{\partial u}{\partial t}+\frac{\partial u}{\partial x} = 3u , u(x,0)=4e^{-x}-e^{-5x} let U =X(x)T(t) so 4X\frac{\partial T}{\partial t}+T\frac{\partial X}{\partial x} = 3XT 4\frac{\partial T}{T \partial t}+\frac{\partial X}{X \partial x} = 3 \left( 4\frac{\partial T}{T...

Homework Statement Let ##\sum_{n=1}^{\infty}a_n## be a series with nonnegative terms which diverges, and let ##(s_n)## be the sequence of partial sums. Prove that ##\lim_{n\to\infty} s_n = \infty##. Homework EquationsThe Attempt at a Solution This isn't a difficult problem, but I want to make...

Homework Statement I have a small question about the following problem. The figure represents the cross-section of a three-conductor system comprising a communications coaxial cable of length l running parallel to a conducting wall (reference conductor). Determine the partial capacitance scheme...

It is mentioned in Reif's book, statistical physics, that trough dimensional analysis it can be shown that: $$\frac{1}{\beta} = kT $$ where ##\beta## equals ##\frac{\partial \ln \Omega}{\partial E}## and k is the Boltzmann constant. I don't quite see how to reach this result, can anyone give me...

Homework Statement What's the difference between Oxygen Tension And Partial Pressure Homework Equations ...uh...rules of grammar ? The Attempt at a Solution If I knew the solution I wouldn't be here now would I ? Here are a few links https://en.m.wikipedia.org/wiki/Blood_gas_tension...

I am integrating the below: \begin{equation} \psi(r,v)=\int \left( \frac{\frac{\partial M(r,v)}{\partial r}}{r-2M(r,v)}\right)dr \end{equation} I am trying to write ψ in terms of M. Please, any assistance will be appreciated.

https://ibb.co/dxwnMe https://ibb.co/miNu1e In these slides they show the partial pressure of the H2O gas not changing when the enternal pressure on the entire gas is increased. Why is this the case? I know it condenses to maintain the same partial pressure, but couldn't the partial pressure of...

Homework Statement Find Partial Fraction Expansion 10/[s (s+2)(s+3)^2] Homework EquationsThe Attempt at a Solution 10/[s (s+2)(s+3)^2] = A/s + B/(s+2) + C/(s+3)^2 + D/(s+3) A = 10/[(s+2)(s+3)^2], s approaches 0 = 10/(2*3^2) = 5/9 B = 10/[s (s+3)^2], s approaches -2 = 10/(-2) = -5 C =...