All Questions
41 questions
1
vote
2
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44
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Perfect gas relation in differential form [closed]
I have a problem to understand the transformation of the perfect gas relation:
$$ \rho\cdot R\cdot T = P $$
into its differential form:
$$\frac {dp}{p} = \frac {d{\rho}}{\rho} + \frac {d{T}}{T}$$
How ...
- 21
0
votes
1
answer
105
views
Derivation of the state equation of a van der Waals gas. Can I invert the derivative to help me?
The state equation of a van der Waals gas is
$$\left(P+\frac{a}{v^2}\right)(v-b)=RT$$
with $a,b$ and $R$ constant. Find $$\frac{\partial v}{\partial T}\bigg\rvert_P.$$
Finding $\frac{\partial v}{\...
- 11
0
votes
1
answer
94
views
What are some ways to derive $\left( \boldsymbol{E}\cdot \boldsymbol{E} \right) \nabla =\frac{1}{2}\nabla \boldsymbol{E}^2$?
For each of the two reference books the constant equations are as follows:
$$
\boldsymbol{E}\times \left( \nabla \times \boldsymbol{E} \right) =-\left( \boldsymbol{E}\cdot \nabla \right) \boldsymbol{E}...
- 105
-1
votes
1
answer
164
views
Given a Postion-time curve/function, how do I find the time spent per unit position?
I have recordings of the position time curve for a given 1D actuator.
I'm trying to find out the time spent per unit length.
To get this relationship, I tried to take an example of a linear function:
$...
- 57
2
votes
1
answer
109
views
$x$-derivative of the wave function and its conjugate [closed]
I saw that in order to show that the normalisability of a wave function does not depend on time, there is a necessary step in the calculation that says that:
$$\left(\Psi^*\frac{\partial^2\Psi}{\...
2
votes
4
answers
261
views
Show that $d\mathbf{v}^2/dt = 2\mathbf{v}\cdot d\mathbf{v}/dt$ using geometry only
I have just begun reading Modern Classical Physics by Thorne and Blandford and I am trying to wrap my head around their "geometric viewpoint" on classical mechanics. The first exercise in ...
- 510
1
vote
1
answer
84
views
Finding back a simple SDE from its solution
I'm trying to self-learn Kurt Jacob's Stochastic Processes for Physicists: Understanding Noisy Systems. I've followed Chapter 3, where I saw how to derive that the solution to the SDE
$$
dx=\left(c+\...
- 11
0
votes
3
answers
82
views
Chain rule when the intermediary variable might be equal to zero
I came across the following question in the kinematics section of my introductory physics textbook:
The velocity of a particle moving along x-axis is given as $v=x^2-5x+4$ (in $m/s$), where $x$ ...
- 31
1
vote
0
answers
187
views
Lienard-Wiechert Potential derivation in Wald's "Advanced Classical Electromagnetism" [closed]
I want to follow the Lienard-Wiechert potential derivation in Robert Wald's E-M book, page 179. I do not understand $dX(t_\text{ret})/dt$ on the right side. I assume the chain rule is applied and $x'^...
- 149
1
vote
1
answer
34
views
Derivatives of the lagrangian of generalized coordinates [closed]
I know that
$$U= \frac{1}{2} \sum_{j,k} A_{jk} q_j q_k \quad \quad T= \frac{1}{2} \sum_{j,k} m_{jk} \dot{q}_j \dot{q}_k $$
and the lagrangian is
$$ \frac{\partial U}{\partial q_k} - \frac{d}{dt} \...
- 11
1
vote
0
answers
167
views
Partial derivatives in thermodynamics: general mathematical procedure [closed]
In the lecture notes (thermodynamics) the following mathematical identity is often used:
$$ \left(\frac{\partial A}{\partial X}\right)_Z = \left(\frac{\partial A}{\partial X}\right)_Y + \left(\frac{\...
- 85
0
votes
1
answer
28
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Clarification for derivatives under a change of variables
In Special Relativity and Classical Field Theory by Susskind, he says that we can imagine a function of $(x+ct)$, then he says that we can consider its derivatives and easily see that $$\frac{\...
- 1,397
-2
votes
1
answer
3k
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What is the General formula of gradient of $r^n$? [closed]
so, the question is that r is the separation vector from a fixed point $(x',y',z')$ to the point $(x,y,z)$ and let $r$ be its length.
the answer to the question of what is the general formula of $$\...
- 17
0
votes
1
answer
61
views
Two questions concerning dirac delta function and Hamiltonian
I'm trying to compute to quantities with Hamiltonian and Dirac delta function but I don't how to do it properly. I'm stuck calculating the following quantity
$$ \frac{d}{dE} \left[ \theta(E-H(x,p;V)) ...
- 2,180
1
vote
2
answers
108
views
Why can we change $dt$ with $(dt/dp)_s dp$?
In my homework assignment there's the following question:
A general thermodynamic system is being compressed isentropically from pressure $P_i$ to $P_f$ while keeping the number of particles constant....
- 97
0
votes
0
answers
36
views
How to evaluate a non-banal derivate?
I need to evaluate the following derivate:
$$\frac{dF}{d\Psi} = \frac{d}{d\Psi}\left[\beta\Delta\Psi+\alpha\left|\Psi\right|^2\Psi+\mu\Psi-i\vec{v}\cdot\bar{\nabla}\Psi\right]$$
where $\Psi$ is a ...
0
votes
3
answers
193
views
Limit of $d\rightarrow 4$ of a function in Peskin & Schroeder
In Peskin & Schroeder section 12.1 equation 12.15 we compute the function
$$
\frac{-3\lambda^2}{(4\pi)^{d/2} \Gamma(\frac{d}{2})}\frac{(1-b^{d-4})}{d-4}\Lambda^{d-4}
$$
Now when we take the limit $...
- 2,083
1
vote
1
answer
86
views
Find that $d\left(\frac{\mu}{T}\right)=ud\left(\frac{1}{T}\right)+vd\left(\frac{P}{T}\right)$
Find that $d\left(\frac{\mu}{T}\right)=ud\left(\frac{1}{T}\right)+vd\left(\frac{P}{T}\right)$
$$U=TS-PV+\mu N\tag{1}\label{1}$$
$$dU=TdS-PdV+\mu d N \tag{2}\label{2}$$
From equation \eqref{1}
$$dU=...
- 17
0
votes
1
answer
81
views
Volume of a two-dimensional sphere in a fixed three sphere geometry
I'm just starting to read Hartle's Gravity and he gives the following equation for the volume of a 2D sphere of radius $r$ if space was a fixed 3-sphere geometry on page 20.
$$V = 4\pi a^3\left(\frac{...
- 5
0
votes
1
answer
60
views
How can I prove this relation between derivatives? [closed]
Consider coaixialcable with TEM. Nonstatic fields are being considered, i.e situation obeys $\nabla \times \mathbf {E}=-\frac{\partial \mathbf{B} }{\partial t} $
If I let a eletric field be described ...
- 1
0
votes
1
answer
33
views
Optimizing a Capacitance function
I am trying to find the optimum values, in order to maximize the following equation:
$$ C (L, (b/a)) =\frac{L 2\pi k\epsilon_0}{\ln(b/a)} $$
where
$$ \frac{dC}{d(b/a)} = -\frac{L2\pi k\epsilon_0 \ln(b/...
- 348
1
vote
1
answer
113
views
Calculating the variation of an operator in two different ways
Let
$$
H_{T}=\dot{x}^{I}\frac{\partial}{\partial \psi^{I}}T(x,\psi)
$$
and consider the transformation:
$$
x^{I}\mapsto x^{I}+i\epsilon\psi^{I}
\\
\psi^{I}\mapsto\psi^{I}-2\epsilon\dot{x}^{I}
$$
where ...
- 587
0
votes
1
answer
435
views
Find the distance travelled between $t=0$ and $t=5$ [closed]
The position vector of a particle is given as $\vec r = \frac43 t^{3/2}\hat i - \frac{1}{2} t^2\hat j + 2 \hat k$, $t$ is in seconds. Find the distance travelled between $t = 0$ and $t = 5$ seconds.
...
- 231
0
votes
2
answers
145
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Spin coherent state path integral derivation
I'm trying to follow the exposition of spin coherent state path integral presented in Condensed Matter Field Theory by Altland and Simons (section 3.3, Page 134-142), and I have a problem with the ...
- 3
0
votes
2
answers
173
views
Taylor Series Expansion of unknown, fraction function
I am learning about deformation, and the deformed state between two points can be defined as
$$E(x) = \frac{(f(x+dx) - f(x))^2 - (dx)^2}{2(dx)^2}$$
My textbook says
When $dx \to 0$ we can use a ...
- 1
8
votes
2
answers
1k
views
What does $\exp\left( ax\frac{d}{dx} \right)$ do on $\psi(x)$?
I'm trying to find out
$$\exp\left(ax\frac{d}{dx}\right)\psi(x)= \ \ ? $$
I tried spending the exponential and then operating the derivatives one by one but I found no pattern. Besides, it gets ...
- 12.1k
-1
votes
1
answer
4k
views
Lennard-Jones potential, distance $r$ for minimum energy
I'm sorry if the question seems stupid. I found (wikipedia) that the Lennard-Jones potential has it's minimum at a distance of
$$r = 2^{\frac{1}{6}}\sigma.$$
If $U(r)_{min} = -\epsilon$
$$U(r) = 4\...
- 137
1
vote
2
answers
89
views
Name this Vector Calculus Theorem
There is an important theorem in vector calculus that says $\boldsymbol{\nabla}\boldsymbol{\cdot}\mathbf{G}\boldsymbol{=}0$
(where $\mathbf{G}$ is some differentiable vector function) implies and is ...
- 409
0
votes
1
answer
27
views
Change of variables for momenta [closed]
http://www.stat.physik.uni-potsdam.de/~pikovsky/teaching/stud_seminar/Wigner_function.pdf
From the Appendix in the above PDF (page 945), below equation (A3) the following expressions are given:
$$
u = ...
- 67
1
vote
1
answer
148
views
Differential form of the velocity equation in non-standard configuration
I'm reading a text on special relativity ($^{\prime\prime}$Core Principles of Special and General Relativity$^{\prime\prime}$, by James H. Luscombe, Edition 2019), in which we start with the equation ...
- 1,071
0
votes
1
answer
31
views
Can someone please explain me how this came? [closed]
I am not getting how above equation is derived using cylindrical coordinates transformations.
This is from page 36, Mathew Sadiku
- 27
1
vote
1
answer
141
views
Taking derivatives of traces over matrix products
I started with evaluating the following derivative with respect to a general element of an $n\times n$ matrix,
$$\frac{\partial}{\partial X_{ab}}\left(\mathrm{Tr}{(XX)}\right)$$
I wrote out the ...
user195631
-1
votes
2
answers
88
views
Change in areal element
Example 1.7
Calculate the surface integral of $\mathbf{v}=2xz\hat{\mathbf{x}}+(x+2)\hat{\mathbf{y}}+y(z^2-3)\hat{\mathbf{z}}$ over five sides (excluding the bottom) of the cubical box (side 2) in Fig. ...
- 1
1
vote
1
answer
177
views
Find $v(t)$ and $x(t)$, How do I treat $δt$? [closed]
We apply a force to a particle with a mass $m$ and inicial velocity $v_0$:
$$ F(t) = \left \{ \begin{matrix} 0 & \mbox{ $t<t_0$}
\\ \frac{p_0}{\delta t} & \mbox{ $t_0<t<t_0 +\...
- 39
-1
votes
2
answers
89
views
For how long is an objects velocity it's instantaneous velocity at time $t$?
Basically I'm asking if an object's instantaneous velocity at time $t$ is $8m/s$ and its instantaneous velocity at time $t^+$ (idk latex, but basically the t + an infinitely small number) is $10m/s$, ...
0
votes
1
answer
2k
views
What is the curl of $k\hat{r}/r^n$?
I'm trying to find the curl of ${\bf F}(r) = k \hat{r}/r^n$. I think that this converts to:
$$
k\left(\frac{\hat{x}}{r} + \frac{\hat{y}}{r} + \frac{\hat{z}}{r}\right)\frac{1}{(x^2 + y^2 + z^2)^{n/2}}
...
- 13
0
votes
1
answer
45
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Equations of motion acceleration doubt
So i was going through some text today morning. Where it said
$$ a = \frac{vdv}{dx} $$
So they then went on to,
$$ vdv = adx \\ \implies \int vdv = \int adx$$
But,I am very certain acceleration is ...
- 23
0
votes
1
answer
58
views
Differential Operator
I am trying to understand the following expression
\begin{eqnarray}
e^{-ik.x}D_{\mu}D^{\mu}e^{ik.x} & = & e^{-ik.x}(i\partial_{\mu}+A_{\mu})(i\partial^{\mu}+A^{\mu})e^{ik.x}\\
& = & e^{...
- 53
1
vote
1
answer
70
views
Derive an equation related to magnetism [closed]
Solve the equations for $v_x$ and $v_y$ :
$$m\frac{d({v_x)}}{dt} = qv_yB \qquad m\frac{d{(v_y)}}{dt} = -qv_xB$$
by differentiating them with respect to time to obtain two equations of the form: $$...
- 119
4
votes
2
answers
18k
views
Why and when do we differentiate or integrate equations in physics? [closed]
I'm an engineering student and none of my professors ever explained why do we use derivations and/or integrations in physics. So I have this task, it goes like:
The object is moving in a positive ...
- 41
2
votes
1
answer
1k
views
Taylor series expansion of $\ln$ and $\cosh$ in distance fallen in time $t$ equation
I want to find the Taylor expansion of $y=\frac {V_t^2}{g} \ln(\cosh(\frac{gt}{V_t}))$
I have tried using the fact $\cosh x= \frac {e^x}{2}$ for large t, which works, I just need help on small values ...
- 345