All Questions
Tagged with differentiation homework-and-exercises
290 questions
1
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113
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Symmetry of the Batalin-Vilkovisky (BV) antibracket operation
Batalin and Vilkovisky define $^1$ an operation they call antibracket which is
$$(F,H)
=
\Big(\frac{\partial_r F}{\partial \Phi^A}\Big)
\Big(\frac{\partial_l H}{\partial \Phi^* _A} \Big)
-
\Big(\frac{...
- 13
1
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1
answer
104
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Where is the error in this calculation of net curl for simple magnetic field?
I wasn't sure whether to post this on MSE, but PSE seems more appropriate.
Let B be a static magnetic field in spherical coordinates, defined as $B=r\hat{\theta}$. Then, it's curl is $$\nabla \times ...
1
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0
answers
50
views
Functional derivative of a symmetrized field
I'm confused whether a symmetrisation/antisymmetrization of a function with respect to its arguments, i.e., $$F(x_1,x_2,...,x_n)=\frac{1}{\sqrt{N!}}\sum_{\pi}\textrm{sgn}(\pi)~f(x_{\pi(1)},...,x_{\pi(...
- 1,363
1
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0
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373
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Second-order covariant derivative in index notation [closed]
So I'm having problems finding the second order covariant derivitive in index notation. My teacher said to just find the covariant derivative of a covariant derivative, so I first started with finding ...
1
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1
answer
671
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Determining the change in radius of water flowing from a faucet - more general question on differentiation
I've been outside of the academic world for several years now, and I'm forcing myself to go back through old textbooks and resources and work through the information in there. I can tell I'm losing ...
- 11
1
vote
1
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130
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Divergence of a specific electrical field [closed]
I need to show that the divergence of the electrical field given as
$$\vec{E}=\vec{e_{\theta}}\frac{A\sin\theta}{r}\exp[i\omega(t-r/c)]$$
is zero. As the vector (in sperical coordinates) containes ...
- 11
1
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1
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120
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The dimensional analysis of the GR geodesic equation
The geodesic equation parametrized by the proper time contains two terms:
$$
{d^{2}x^{\mu } \over ds^{2}}=-\Gamma ^{\mu }{}_{{\alpha \beta }}{dx^{\alpha } \over ds}{dx^{\beta } \over ds}\
$$
The ...
1
vote
1
answer
70
views
Derive an equation related to magnetism [closed]
Solve the equations for $v_x$ and $v_y$ :
$$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
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3
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193
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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
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3
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260
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How to show that $\nabla \times \vec{r}/{|\vec{r}|} = 0$? [closed]
That's basically it:
$$\nabla \times \frac{\vec{r}}{|\vec{r}|} = 0$$
There's some connection with physical significance of curl of vector function. I am currently in Grade 10 but am in a preparatory ...
0
votes
2
answers
173
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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
0
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3
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3k
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Proof divergence of magnetic field is 0
I work in an R&D role that involves magnetism. I am refreshing my memory of electromagnetic and this stumps me. In polar coordinates, the magnetic field of a current loop for distances $R >>...
- 23
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1
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238
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Why do these equations result an incorrect unit for acceleration?
Hello everyone.
Imagine an object moving around a certain point on a circular orbit. Magnitude of the velocity is constant during the motion ($|v|$). The orbit radius is $r$. (I'd better notice that ...
- 1,914
0
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2
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164
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Derivation of $x \partial_y - y\partial_x = \partial_{\phi}$
On a $S^2$-sphere we can define the coordinates $$x = \sin(\theta)\cos(\phi)\\ y = \sin(\theta)\sin(\phi)\\z=\cos(\theta).$$ Then I want to show that $$x \partial_y - y\partial_x = \partial_{\phi}.$$
...
- 147
0
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1
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326
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Momentum operator dot a vector
Why is $P \cdot A = A \cdot P -i\hbar\nabla \cdot A$? I was just replacing $P=-i\hbar\nabla $ so I didn't get the first term on the right side
0
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1
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94
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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:
$$
\boldsymbol{E}\times \left( \nabla \times \boldsymbol{E} \right) =-\left( \boldsymbol{E}\cdot \nabla \right) \boldsymbol{E}...
- 105
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1
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82
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Simultaneously raising and lowering indices
Let $U$ be a four-vector and $\nabla$ denote the covariant derivative in the Levi-Civita connection. Is it always true that $$\left(\nabla_{\mu}U^{\nu}\right)U_{\nu}=\left(\nabla_{\mu}U_{\nu}\right)U^{...
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1
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204
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Can you apply product rule to arg of a bra-ket?
I found the following expression in a paper:
$$
\frac{d}{dt}\arg\langle\phi_+|\dot{\phi_-}\rangle
$$
where the $\arg$ term is the argument of the complex number given by inner product between two ...
- 591
0
votes
1
answer
74
views
Tensor Index Manipulation
I am trying to study General Relativity and I thought about starting with some index gymnastics. I found a worksheet online and I am stuck with a simple problem. I have to prove that
$$\partial_{\mu} ...
- 1
0
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2
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329
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Transformation of Lie derivative of one-form
In the textbook Supergravity ( by Freedman and Proeyen, 2012), they have defined the Lie derivative of a covariant vector with respect to a vector field V on page 139:
$$ \mathcal{L}_V \omega_\mu = V^\...
- 542
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1
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162
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How does one calculate partial derivatives with two constant variables in statistical mechanics
I came across this relation which I have yet to be able to prove or find proof of:
$$kT^2\left(\frac{\partial \ln\mathscr{Z}}{\partial T}\right)_{V,\mu}=\langle H\rangle-\mu\langle N\rangle$$
I was ...
- 23
0
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2
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609
views
How does a charged particle behave in a vector potential?
I know that a charged particle interacts with a magnetic field through the Lorentz force, thus knowing how it behaves in a given magnetic field.
However, I don't understand how a charged particle (be ...
0
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3
answers
779
views
How to take the curl of the angular momentum operator?
I'm trying to show $$\nabla \times \vec{L} = \frac{1}{i}(\vec{r}\nabla^{2}-\nabla(1+r\frac{\partial}{\partial{r}})) $$ where $ \vec{L}\psi = \frac{1}{i}(\vec{r}\times \nabla)\psi$.
I'm able to expand ...
- 11
0
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1
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197
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Show that $\nabla ·(\bar\Psi\nabla\Psi - \Psi\nabla\bar\Psi) = \bar\Psi\nabla^2\Psi - \Psi\nabla^2\bar\Psi$ [closed]
Show that
$$\nabla ·(\bar\Psi\nabla\Psi - \Psi\nabla\bar\Psi) = \bar\Psi\nabla^2\Psi - \Psi\nabla^2\bar\Psi$$
where $\Psi$ is a complex wave equation, $\nabla$ is the gradient and $\bar\Psi$ is the ...
- 221
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1
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274
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Kinetic energy derivation: Why is $\frac{d \mathbf v}{dt} \cdot \mathbf v= \frac 12 \frac{d}{dt}(v^2)~?$
In Goldstein's Classical Mechanics 3rd edition, page 3, the Kinetic energy is derived by considering the work done on a particle by an external force $\mathbf F$ from point $1$ to point $2$ $$W_{12}=\...
user138458
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1
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353
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Geodesic equation proof confusing me
I was looking through this proof and have no idea where the $u$ comes from. Any help is appreciated.
This is from here; I want to know how they got from eqn 5 to eqn 6.
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1
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2k
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Covariant derivative of stress-energy tensor for a scalar field [closed]
In order to prove that $$\nabla ^\mu T_{\mu\nu} =0$$ I want to find the
covariant derivative of $$T_{\mu\nu} = \partial_\mu\phi \partial_\nu \phi -\frac{1}{2}g_{\mu\nu}(g ^{\lambda\sigma}\partial_\...
- 1,539
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2
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59
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Help with Commutators [closed]
I'm trying to self study quantum mechanics and am having a little trouble manipulating commutators. I get two different answers below, depending on the method I'm using. The second method gives me the ...
- 81
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1
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53
views
What did I do wrong? I got $\nabla\cdot \vec A \neq div \vec A $ [closed]
We know, that in orthogonal Curvilinear coordinate system:
$$ \nabla =\sum_{i=1} ^{3}{\hat{e_i} \over h_i}{\partial \over\partial u_i} $$
Let
$$\vec A=\sum_{i=1} ^{3} A_i \hat e_i$$
Now
$$ \nabla ...
- 123
0
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1
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425
views
Taylor expansion of scalar fields [closed]
Starting of with electrodynamics I have to compute the taylor expansion around $\vec{r} = 0$ of
$\psi (\vec{r}) = |\vec{r} - \vec{r_0}|^{\frac{3}{2}}$ where $\vec{r_0}$ is a constant vector up to ...
- 249
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1
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105
views
How to expand $(D_\mu\Phi)^\dagger(D^\mu\Phi)$ in $SU(2)$?
I would like to calculate the following expression:
$$(D_\mu\Phi)^\dagger(D^\mu\Phi)$$ where $$D_\mu\Phi = (\partial_\mu-\frac{ig}{2}\tau^aA_\mu^a)\Phi$$ and $A_\mu^a$ are the components of a real $SU(...
- 53
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1
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Clarification for derivatives under a change of variables
In Special Relativity and Classical Field Theory by Susskind, he says that we can imagine a function of $(x+ct)$, then he says that we can consider its derivatives and easily see that $$\frac{\...
- 1,397
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1
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33
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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
$$ \frac{dC}{d(b/a)} = -\frac{L2\pi k\epsilon_0 \ln(b/...
- 348
0
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1
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435
views
Find the distance travelled between $t=0$ and $t=5$ [closed]
The position vector of a particle is given as $\vec r = \frac43 t^{3/2}\hat i - \frac{1}{2} t^2\hat j + 2 \hat k$, $t$ is in seconds. Find the distance travelled between $t = 0$ and $t = 5$ seconds.
...
- 231
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2
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397
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Vector identity of the convective term of Navier-Stokes equation
In the NS, a well known expression of the convective term is
$$\bf v \times (\nabla\times \bf v) = \bf v\cdot \nabla v - \frac{1}{2}\nabla v^2 $$
In order to derive it I use the commute rule of the ...
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1
answer
1k
views
Proof that covariant derivative of contravariant components of metric vanish for metric compatabilit
In this excercise I want to show that $\nabla_\rho g_{\mu \nu}=0$ and $\nabla_\rho g^{\mu \nu}=0$
This should probably be very easy, but excuse me I'm completly new to GR.
So to do this I used that ...
- 153
0
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3
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977
views
Verify that the electrostatic potential satisfies the Poisson equation [closed]
I'm reading Sect1.7 of Jackson's classical electrodynamics but I have trouble following his argument. Could someone help explain how exactly the Laplacian is evaluated in 1.30? Is it calculated with ...
- 9
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3
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185
views
Proof for Vector Identity
I am currently studying electrodynamic and came across the following vectoridentity, but I am unable to prove it:
$$ \vec{f} \times ( \nabla \times \vec{f} ) -\vec{f}(\nabla\cdot\vec{f}) = \nabla \...
- 3
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1
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399
views
Ambiguous curl for a vector field
I'm trying to work out how to explain why the curl/divergence for the following image is ambiguous without giving the explicit vector function for the following:
E has an ambiguous divergence and I ...
- 3,240
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4
answers
5k
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Sign of acceleration from position-time graph
These three graphs are from my textbook. It states that the acceleration in 1) is positive, 2) is negative and 3) is zero and can be told by looking at the slope.
What I understand from the graph is ...
- 493
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1
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108
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Acceleration varies inversely with 3rd power of displacement
Question. A particle is moving in a straight line. Displacement $x$ and time $t$ of the particle are related by the equation
$$x^2=at^2+2bt+c~;~\text{where }a,b,c\text{ are constants.}$$
Prove ...
0
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2
answers
164
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Problem with derivatives for spherical coordinates [closed]
I got stuck with a derivative. I can't think of a solution for this, because I am taking the derivative of a function with respect to its integral. Theta and phi are generalized coordinates. I am ...
- 143
0
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1
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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
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2
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148
views
Calculate divergence via partial derivative [closed]
I need to calculate the divergence and curl for a vectorfield. I've done that before so that's no problem :) Or I've done it using partial derivative, maybe there are multiple ways to solve for ...
- 35
0
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1
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91
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In central-force mechanics, how do we substitute $ξ=\frac{1}{r}$?
I have taken a look at central-force mechanics in the past, but I still cannot understand how $ξ=\frac{1}{r}$ is substituted to find $\frac{d^2r}{dt^2}$ in terms of ξ.
So I know from $F=ma$ that:
$$(...
- 21
0
votes
1
answer
123
views
How do I set when the object isn't moving
I started studying instantaneous velocity derivatives using only now.
It may seem stupid but really I'm not sure whether that's right: I have an equation:
$$x (t) = 1.5t - 9,75t³$$
To set the time ...
- 107
0
votes
2
answers
1k
views
Divergence of vector potential [closed]
I was given the vector potential $$\vec A (\vec r) = - \vec a \times \nabla \frac{1}{r}$$ with a constant vector $\vec a$. Now, I found the $\vec B$ field which is I think $- \vec a \frac{2}{r^3}$, ...
- 4,830
0
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1
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65
views
Maximal Velocity of an Object in Free Fall
A baseball is dropped from a high point. Since the velocity is large, we can say that the drag force is proportional to the square of the velocity, $F_d = \gamma v^2$. My goal is to determine the ...
- 230
0
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1
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268
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Gradient of two-particle system
I'm working on problem 5.1a from Griffiths Intro to QM and given that:
$$\mathbf R \equiv \frac{m_1\mathbf{r_1} + m_2 \bf r_2}{m_1+m_2}$$
and $\bf r \equiv \bf r_1 - \bf r_2$ I need to show that,
$$\...
- 207
0
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1
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105
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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