All Questions
Tagged with differentiation homework-and-exercises
290 questions
5
votes
4
answers
386
views
Vector triple product with $\nabla$ operator
I came across the following expression in several books (especially in plasma physics literature while deriving the magnetic pressure):
$$(\mathbf{\nabla} \times \mathbf{B})\times \mathbf{B} = \left(\...
- 61
1
vote
2
answers
44
views
Perfect gas relation in differential form [closed]
I have a problem to understand the transformation of the perfect gas relation:
$$ \rho\cdot R\cdot T = P $$
into its differential form:
$$\frac {dp}{p} = \frac {d{\rho}}{\rho} + \frac {d{T}}{T}$$
How ...
- 21
0
votes
1
answer
105
views
How to expand $(D_\mu\Phi)^\dagger(D^\mu\Phi)$ in $SU(2)$?
I would like to calculate the following expression:
$$(D_\mu\Phi)^\dagger(D^\mu\Phi)$$ where $$D_\mu\Phi = (\partial_\mu-\frac{ig}{2}\tau^aA_\mu^a)\Phi$$ and $A_\mu^a$ are the components of a real $SU(...
- 53
1
vote
2
answers
105
views
Why must a constraint force be normal?
If we impose that a particle follows a holonomic constraint, so that it always remains on a surface defined by some function $f(x_1,x_2,x_3)=0$ with $f:\mathbb{R^3}\rightarrow\mathbb{R}$, we get a ...
- 131
1
vote
2
answers
325
views
Question regarding error analysis of focal length of a lens [duplicate]
The question in whose context i am asking this question is as follows
In an experiment for determination of the focal length of a thin convex lens, the distance of the object from the lens is $10 \pm ...
- 173
0
votes
1
answer
105
views
Derivation of the state equation of a van der Waals gas. Can I invert the derivative to help me?
The state equation of a van der Waals gas is
$$\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.$$
Finding $\frac{\partial v}{\...
- 11
0
votes
1
answer
69
views
D'Alembert Solution to 1+1D wave equation - integration step
I am working through d'Alembert's solution to the 1+1D wave equation using the substitution of canonical coordinates. I have an initial condition of: $$u_{t}(x,0) = g(x) $$ with a general solution ...
1
vote
1
answer
109
views
Spherical coordinate of a vector when divergence of the vector is zero
$\nabla \cdot \mathbf{\delta u_{perp}} = 0$ where $\mathbf{\delta u_{perp}}$ is a function of both x and y coordinates and perpendicular to z axis. Moreover, $\delta u_{perp}$ along z axis is $0$.
I ...
- 31
-3
votes
1
answer
69
views
Show that $dE/dt = -bv^2$ (Help with differentiation) [closed]
The question is:
Show that $$dE/dt = -b (dx/dt)^2.$$
And the solution is:
...
- 1
3
votes
3
answers
116
views
Finding the vector potential
$$\nabla\times\mathbf{B}=\nabla\times\left(\nabla\times\mathbf{A}\right)=\nabla\left(\nabla\cdot\mathbf{A}\right)-\nabla^2\mathbf{A}=\mu_0\mathbf{J}\tag{5.62}$$
Whenever I try to work this out and ...
- 220
1
vote
2
answers
117
views
Derivative of displacement in deriving expression for intensity of sound waves
I am currently working on deriving the expression for intensity of a sound wave: https://openstax.org/books/university-physics-volume-1/pages/17-3-sound-intensity
The previously mentioned book states: ...
- 19
0
votes
1
answer
94
views
What are some ways to derive $\left( \boldsymbol{E}\cdot \boldsymbol{E} \right) \nabla =\frac{1}{2}\nabla \boldsymbol{E}^2$?
For each of the two reference books the constant equations are as follows:
$$
\boldsymbol{E}\times \left( \nabla \times \boldsymbol{E} \right) =-\left( \boldsymbol{E}\cdot \nabla \right) \boldsymbol{E}...
- 105
2
votes
2
answers
152
views
How to calculate the rotation at a singularity?
An electrodynamics lecture asks me to prove that
$$
\nabla \times \left( \frac{\vec{M} \times \vec{x}}{ |\vec{x}|^3} \right) = \frac{8 \pi}{3} \vec{M} \delta^3(\vec{x})- \frac{\vec{M}}{|\vec{x}|^3}+ \...
- 151
0
votes
1
answer
74
views
Tensor Index Manipulation
I am trying to study General Relativity and I thought about starting with some index gymnastics. I found a worksheet online and I am stuck with a simple problem. I have to prove that
$$\partial_{\mu} ...
- 1
-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:
$$\left(\Psi^*\frac{\partial^2\Psi}{\...
0
votes
2
answers
329
views
Transformation of Lie derivative of one-form
In the textbook Supergravity ( by Freedman and Proeyen, 2012), they have defined the Lie derivative of a covariant vector with respect to a vector field V on page 139:
$$ \mathcal{L}_V \omega_\mu = V^\...
- 542
4
votes
2
answers
641
views
Confusion on metric determinant derivative
Maybe it is a stupid confusion. I need to compute the derivative of the metric determinant with respect to the metric itself, i.e., $\partial g/\partial g_{\mu\nu}$, but I have an indices confusion in ...
- 378
0
votes
2
answers
59
views
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
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
$$
dx=\left(c+\...
- 11
0
votes
3
answers
82
views
Chain rule when the intermediary variable might be equal to zero
I came across the following question in the kinematics section of my introductory physics textbook:
The velocity of a particle moving along x-axis is given as $v=x^2-5x+4$ (in $m/s$), where $x$ ...
- 31
1
vote
3
answers
64
views
Showing that intensive parameters obtain by considering molar quantities
In Callen's Thermodynamics textbook, he writes that
$$\left(\frac{\partial u}{\partial s}\right)_v = \left(\frac{\partial U}{\partial S}\right)_{V,N}$$
where $u = U/N$, $s = s/N$, and $v = V/N$ and, ...
- 1,271
0
votes
1
answer
82
views
Simultaneously raising and lowering indices
Let $U$ be a four-vector and $\nabla$ denote the covariant derivative in the Levi-Civita connection. Is it always true that $$\left(\nabla_{\mu}U^{\nu}\right)U_{\nu}=\left(\nabla_{\mu}U_{\nu}\right)U^{...
1
vote
2
answers
286
views
What is the Laplacian of $k\hat{r}$ where $r=\sqrt{x^2+y^2}$ and $k$ is a constant? [closed]
Using 3D cylindrical coordinates, I get 0 as the answer.
$$ \nabla^{2} (k \hat{r}) = \hat{r} (\frac{1}{r} \frac{\partial }{\partial r}\left(r \frac{\partial (k)}{\partial r}\right) + 0 + 0) + \hat{\...
- 19
0
votes
0
answers
72
views
A calculation question about Taylor expansion in Altland and Simons p 106, the gutter like potential
I have a question regarding the book condensed matter field theory by Altland Simons p 106. In their a gutter like potential is given and it is required to calculate the fluctuation $\delta V_{tension}...
- 37
3
votes
1
answer
238
views
Is the time derivative of the adjoint equal to the adjoint of the time derivative?
This is hopefully straightforward. Starting from the Schrödinger equation as an axiom, one obtains the operator differential equation for the $U$ such that $| \psi(t) \rangle = U(t,t_0) | \psi(t_0) \...
- 1,271
4
votes
4
answers
198
views
How to prove $\mathrm{Tr}[(\partial_\mu U)U^\dagger]=0$?
I am studying ChPT by referring to "A Primer for Chiral Perturbation Theory" by Stefan Scherer.
I'm having a problem with the consideration of terms that appear in the Lagrangian.
The ...
- 173
1
vote
0
answers
187
views
Lienard-Wiechert Potential derivation in Wald's "Advanced Classical Electromagnetism" [closed]
I want to follow the Lienard-Wiechert potential derivation in Robert Wald's E-M book, page 179. I do not understand $dX(t_\text{ret})/dt$ on the right side. I assume the chain rule is applied and $x'^...
- 149
1
vote
1
answer
34
views
Derivatives of the lagrangian of generalized coordinates [closed]
I know that
$$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
votes
1
answer
51
views
Proving the relation $\frac 1 2 \left[\nabla^2,r \right] = \frac 1 r + \frac \partial {\partial r}$ (quantum mechanics exercise) [closed]
I'm trying to prove this relation in my quantum mechanics exercise book
$$\frac 1 2 \left[\nabla^2,r \right] = \frac 1 r + \frac \partial {\partial r}.$$
Here's my attempt:
Expand the Laplacian ...
- 301
1
vote
1
answer
115
views
Divergence not defined
I’m currently working on the practice problems in Introduction to Electrodynamics by Griffiths. I got confused by the solution to this problem.
What does “ill-defined divergence” even mean? I ...
- 353
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:
$$ \left(\frac{\partial A}{\partial X}\right)_Z = \left(\frac{\partial A}{\partial X}\right)_Y + \left(\frac{\...
- 85
4
votes
1
answer
230
views
Is there a quick way to calculate the derivative of a quantity that uses Einstein's summation convention?
Consider $F_{\mu\nu}=\partial_{\mu}A_\nu-\partial_\nu A_\mu$, I am trying to understand how to fast calculate $$\frac{\partial(F_{\mu\nu}F^{\mu\nu})}{\partial (\partial_\alpha A_\beta)}$$
without ...
- 862
1
vote
1
answer
77
views
Thermodynamic Identity
I want to show the following thermodynamic identity (Pathria, 3rd Edition, Appendix H, Pg 677):
$$\left(\frac{\partial x}{\partial y}\right)_w = \left(\frac{\partial x}{\partial y}\right)_z + \left(\...
- 2,740
0
votes
1
answer
162
views
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
votes
1
answer
28
views
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 $$\frac{\...
- 1,397
-2
votes
1
answer
3k
views
What is the General formula of gradient of $r^n$? [closed]
so, the question is that r is the separation vector from a fixed point $(x',y',z')$ to the point $(x,y,z)$ and let $r$ be its length.
the answer to the question of what is the general formula of $$\...
- 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
$$ \frac{d}{dE} \left[ \theta(E-H(x,p;V)) ...
- 2,180
0
votes
1
answer
155
views
Finding the Euler-Lagrange equation for a scalar field
Consider a scalar field with the following lagrangian density:
$$\mathscr{L}=-\frac{1}{2} \partial_{\mu} \phi \partial^{\mu} \phi-V(\phi).$$
I want to find the corresponding Euler-Lagrange equation, ...
- 69
1
vote
2
answers
108
views
Why can we change $dt$ with $(dt/dp)_s dp$?
In my homework assignment there's the following question:
A general thermodynamic system is being compressed isentropically from pressure $P_i$ to $P_f$ while keeping the number of particles constant....
- 97
0
votes
0
answers
36
views
How to evaluate a non-banal derivate?
I need to evaluate the following derivate:
$$\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]$$
where $\Psi$ is a ...
2
votes
3
answers
198
views
Derivation of entropy, I don't understand the relation $ \frac{\partial S_2}{\partial E_1} = -\frac{\partial S_2}{\partial E_2} $
My course guide gives the following derivation for change in entropy w.r.t. energy, where I don't understand a step:
\begin{align}
E & = E_1 + E_2 \\
S & = S_1 + S_2 \\
S(E,E_1 ) & = S_1 (...
- 2,180
4
votes
3
answers
354
views
Ricci Identity with Torsion Proof
In the notes I'm using for General Relativity, the author begins their proof of the Ricci identity with torsion by writing
$$\nabla_{[\mu}\nabla_{\nu]}Z^{\sigma}=\partial_{[\mu}(\nabla_{\nu]}Z^{\sigma}...
1
vote
1
answer
58
views
Energy change under point transformation
How do the energy and generalized momenta change under the following
coordinate
transformation $$q= f(Q,t).$$
The new momenta: $$P = \partial L / \partial \dot Q = \partial L / \partial \dot q\times ...
- 990
1
vote
1
answer
246
views
Four-vector differentiation (E-M Euler-Lagrange eq.)
$$\partial_{\mu} \frac{\partial(\partial_{\alpha}A_{\alpha})^2}{\partial(\partial_{\mu}A_{\nu})} = \partial_{\mu}\left[2(\partial_{\alpha}A_{\alpha})\frac{\partial(\partial_{\beta}A_{\gamma})}{\...
- 149
-3
votes
2
answers
118
views
Explain this equation mathematically
$$\Bigl( \frac{\partial S}{\partial T} \Bigr)_H = \Bigl( \frac{\partial S}{\partial T} \Bigr)_M + \Bigl( \frac{\partial S}{\partial M} \Bigr)_T \Bigl( \frac{\partial M}{\partial T} \Bigr)_H$$
How can ...
- 446
4
votes
5
answers
1k
views
Problem understanding the use of differentiation
I am new to differentiation. Our physics teacher gave us this example problem:
The radius of a sphere is continuously increasing at the rate of 1 m/sec. Find the rate of change of the volume of the ...
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
103
views
I need help with a commutator! [closed]
In QM class this morning my Prof claimed that the commutator $$[𝑥(𝜕/𝜕𝑦), 𝑦(𝜕/𝜕𝑥)] = 0.$$
However, my classmate and I arrived at $$x(𝜕/𝜕x) - y(𝜕/𝜕y).$$
Our professor used the identity $$[𝜕...
