All Questions
Tagged with differentiation homework-and-exercises
290 questions
2
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244
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Is Goldstein's matrix formalism to Hamiltonian mechanics necessary? [closed]
I am trying to see whether the matrix formalism of the Hamiltonian formalism (used in Goldstein's textbook) is truly necessary to solve problem in this framework.
It appears so based on the problem I'...
- 1,081
2
votes
2
answers
88
views
Why does my answer vary?
Q)A wave moves with speed 300 m/s on a wire which is under the tension of 500N.Find how much tension must be changed to increase the speed to 312m/s.
My method: Since $v= \sqrt{T/μ}.....(i)$,where T ...
- 145
2
votes
1
answer
535
views
Expectation value of derivative of operator
I was given the following question:
Let $A(\lambda)$ be a Hermitian operator, which is dependent on some real parameter $\lambda$. Let us denote the eigenvalues and corresponding eigenstates of $A$ ...
- 353
2
votes
1
answer
161
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$\mathbf{g}(\mathbf{r})=-\boldsymbol{\nabla}\psi(\mathbf r)$: searching for a minus sign error
Consider the following figure
where $R=\sqrt{(x-x')^2+(y-y')^2+(z-z')^2}=|\mathbf{r}-\mathbf{r}'|$ is the module of the $\mathbf{R}$ vector depends not only on the location of the $P$ point but also ...
- 2,575
2
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1
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124
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Can these two terms cancel out?
In trying to prove that $$\Gamma_{\mu\nu\lambda} = \eta_{ab}J^a_bJ^b_{\nu\lambda}.$$
The author canceled out while expanding the first equation $$J^a_{\mu\lambda}J^b_\nu$$ with $$J^b_{\mu\lambda}J^a_\...
- 1,539
2
votes
2
answers
152
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How to calculate the rotation at a singularity?
An electrodynamics lecture asks me to prove that
$$
\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
2
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1
answer
58
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Complex Tensors and Metric [closed]
It has been given that in 2-dimensions, consider the metric: $$\ ds^2 = e^{\phi(z,\overline z)}dzd\overline z .$$
Show that $$\ t_{z...z;z} = (\partial_{z} - n\partial_{z} \phi)t_{z....} .$$
I can't ...
- 23
2
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1
answer
215
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Tensor Differentiation
In the book "Tensors, Relativity and Cosmology" the author derived Maxwell's Equation in covariant form using the EM field tensor Lagrangian $L=-\frac{1}{4}F^{jl}F_{jl}$ (source=0). One of the steps ...
- 1,047
2
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1
answer
838
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Galilean transformation and differentiation
Given $x=x’-vt$ and $t=t’$, why is $\frac{\partial t}{\partial x’}=0$ instead of $1/v$? $t$ seems to depend on $x’$ because if $t$ changes, $x’$ changes. Also, in this problem, $dx=dx’$ as well, but I ...
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2
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2
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627
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What conditions are required for the derivative of kinetic energy to be F.v?
In Ch. 1 Derivation 1 of Goldstein's mechanics, we have
Show that for a single particle with constant mass the equation of motion implies
$$
\frac{dT}{dt} = \vec{F}\cdot\vec{v}
$$
The first step ...
- 647
2
votes
1
answer
288
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Lie derivative in this paper [closed]
Say, $$L_{V}z^A =0$$ but I don't know much about Lie derivatives except what I saw now through wikipedia http://en.wikipedia.org/wiki/Lie_bracket_of_vector_fields#Definitions that it is (if I am ...
- 1,539
2
votes
1
answer
2k
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Help deriving the general linear wave equation $d^2y/dx^2=(1/v^2)d^2y/dt^2$ [closed]
How do I derive the General Linear Wave Equation $$d^2y/dx^2=(1/v^2)d^2y/dt^2?$$
My teacher differentiated the general wave function $f(x + vt)+g(x - vt)$ twice with respect to both variables to get ...
- 33
2
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1
answer
598
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Clarification on a Goldstein formula steps (classical mechanics)
At page 20 of Classical Mechanics' Goldstein (Third edition), there are these two steps given between eqs. (1.51) and (1.52):
$$\sum_i m_i \ddot {\bf r}_i \cdot \frac{\partial {\bf r_i}}{ \partial ...
- 1,143
2
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1
answer
872
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Partial Legendre transform: understanding a simple example
Consider the following function:
$$f(x_1, x_2) = x_1^2x_2+x_1x_2^3$$
$f$ is a function of $(x_1, x_2)$. The conjugate variables $(u_1, u_2)$ to $(x_1, x_2)$ are $$u_1 = \partial f/ \partial x_1 = 2 ...
- 1,026
2
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1
answer
726
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How does one deal with derivative operator in quantum field theory properly?
Given creation and annihilation operators, ${a^{\dagger}(x,t)}$ and $a(x,t)$ in non-relativistic quantum field theory, respectively, which satisfy the following properties:
Now, I want to prove $$[...
2
votes
1
answer
384
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Partial derivatives of chiral superfields with respect to scalar components in the superpotential
I am following some notes on supersymmetry by Matteo Bertolini and I need some clarification.
Chapter five deals with sypersymmetric Lagrangians and the superpotential is introduced. It is stated ...
- 6,137
2
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1
answer
122
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a problem on finding acceleration by differentiation
The displacement of particle along the $x$ and $y$ axis is
\begin{cases}
x(t)=\omega t-\sin\omega t\\
y(u)=1-\cos\omega t
\end{cases}
Upon differentiation, the velocity is
\begin{cases}
v_x(t)=\omega\...
2
votes
2
answers
346
views
Determining Acceleration Based On Graph
I understand how to solve this problem, but I am unsure how to generate an equation for the graph (below). My current attempt involves using the mass provided along with the derivative of the line (...
- 123
2
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1
answer
1k
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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
2
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3
answers
504
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About field gradient
I read the term field gradient in most of the article about magnetic field. I search it online but most of the explanation is about the math. I wonder in physics, what the gradient field really mean? ...
- 2,383
2
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1
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11k
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Gravitational force exerted by a rod on a point mass
I have doubts with the solution of a certain problem. I will give the entire solution below and will lay out my doubts as well.
A point mass $m_1$ is separated by a distance $r$ from a long rod of ...
- 383
2
votes
1
answer
154
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How obtain the last expression of the Killing equation?
In order to write down the Killing equation, if by definition a vector field $X$ is said to be Killing $\iff$ $L_X g=0$, then I can rewrite this condition as:
$$L_X g=X g(U, V)-g(L_XU, V)-g(U,L_XV)=g(\...
- 133
2
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0
answers
345
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Supercovariant Derivative action
My query is with Weinberg Vol3 equation just above 26.7.22
Weinberg follows Majorana Superfield formalism. Where, covariant derivative is defined as,
$$D_{R\alpha}=-\epsilon_{\alpha \beta}\frac{\...
- 142
2
votes
1
answer
48
views
Confirmation of Uncertainty in Indices New Formula? [closed]
I am experimenting relations with regards of the value with uncertainty raised to the $n$th power.
I came up with this formula: $$(A\pm\alpha)^n=A^n\pm(A^{n-1}n\alpha)$$
Anyone here able to ...
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2
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0
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184
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Problems while doing $\dfrac{\partial}{\partial(\partial_\mu \phi)}$ and $\dfrac{\partial}{\partial(\partial_\mu A_\mu)}$
In David Tong's lectures, he gives two Lagrangians as examples to derive the equations of motion:
$$\mathcal{L} = \dfrac{1}{2}\eta^{\mu\nu}\,\partial_\mu\phi\,\partial_\nu \phi-\dfrac{1}{2}m^2\phi^2, ...
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1
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4
answers
3k
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What is proper time, proper velocity and proper acceleration?
I am trying to derive the relativistic rocket equations found here [(4),(5),(6),(7),(8)] but I do not understand proper time, proper velocity and proper acceleration.
Define a point $P$ with ...
- 201
1
vote
4
answers
9k
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Dot product of vector and its derivative with respect to time? How does $L \cdot\frac{dL}{dt} = \frac{1}{2}\frac{d(L^2)}{dt}$? [closed]
How does:
$$L \cdot\frac{dL}{dt} = \frac{1}{2}\frac{d(L^2)}{dt}$$
where L is a vector (I dunno how to make it bold in the equation).
How do they reach to this right hand side equation?
And what is ...
- 143
1
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2
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117
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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
1
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3
answers
245
views
Apparent dimensional mismatch after taking derivative
Suppose I have a variable $x$ and a constant $a$, each having the dimension of length. That is $[x]=[a]=[L]$ where square brackets denote the dimension of the physical quantity contained within them.
...
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1
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3
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4k
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Which quantity gives the resistance of a component?
In a current vs potential difference graph, we can obtain the value of the resistance of the component. There are books that say gradient-inverse is the resistance and also books that say the value of ...
- 146
1
vote
1
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378
views
Covariant derivative of spherical harmonics
Given is the metric $\gamma_{jk}$ for the surface of a Sphere $S^2$ with $\gamma_{22}=1,\gamma_{23}=\gamma_{32}=0$ and $\gamma_{33}=\sin^2(\theta)$. The coordinates are $x=$($t,r,\theta,\phi$) and $j$ ...
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1
vote
2
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4k
views
Derivative of the magnetic field to the vector potential
So the magnetic field is defined with the vector potential A as:
$$\mathbf{B}=\nabla\times\mathbf{A}.$$
How would I calculate the derivative:
$$\frac{\delta}{\delta\mathbf{A}}|\mathbf{B}|^2$$
I ...
- 3,132
1
vote
2
answers
371
views
Why is $\nabla\cdot(\hat{\bf r}/r^2)$ giving 0 as answer? [closed]
While I was reading I encountered the statement $\nabla\cdot(\hat{\bf r}/r^2)$ (r cap divided by $r$ square) is 0. Can anyone explain proof of the statement why is it giving 0?
- 21
1
vote
2
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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
1
vote
2
answers
819
views
Showing the form of the covariant derivative of $\phi$, if $\phi$ transforms as the adjoint representation of $SU(n)$
I want to show that if $\phi$ transforms as the adjoint representation of SU(n), its covariant derivative is given by $\textbf{D}_\mu \phi = \partial_\mu \phi + i [\textbf{A}_\mu, \phi]$. (Exercise in ...
- 1,055
1
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3
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143
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Passing from curl to vector product
I don't understand how to obtain second equation with first part in the equation
$$
\nabla \times \vec A_0 e^{-j \vec k\cdot \vec r} = -j\vec k\times \vec A_0 e^{-j \vec k\cdot \vec r}.
$$
Can you ...
- 13
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
1
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1
answer
107
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Show that $\partial_i A_j - \partial_j A_i = \epsilon_{ijk}B_k$
Let us start from $\textbf{B}=\nabla \times \textbf{A}$ and write its components $B_k=\epsilon_{ijk}\partial_i A_j$.
I want to show that $\partial_i A_j - \partial_j A_i = \epsilon_{ijk}B_k$. I can ...
- 551
1
vote
1
answer
511
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Derivation of gradient of the expectation of local energy
Background: In Variational Monte Carlo, given a Hamiltonian $H$ and a wave function $\psi_\alpha$ dependent on some parameter(s) $\alpha$, we have defined a quantity known as the local energy,
$$E_L =...
- 263
1
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1
answer
317
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Thermodynamics - please check my proof that $\partial C_p/\partial p$ = 0 for an ideal gas
Prove
$$\left(\frac{\partial C_p}{\partial p}\right)_T = 0$$ for an ideal gas.
All the $\partial$s are partial derivatives
Please check to see if this makes sense.
We know that
$$C_p = \left(\...
- 137
1
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2
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160
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Why $\sum\limits_{i} \frac{\partial L}{\partial \dot{q_i}} \dot{q_i} = \sum\limits_{i} \frac{\partial T}{\partial \dot{q_i}} \dot{q_i} = 2T$? [closed]
From Landau and Lifschitz's "Mechanics"; section 6.
I understand up to this point
$$E \equiv \sum\limits_{i} \dot{q_i}\frac{\partial L}{\partial \dot{q_i}} - L $$
Then the author states:
Using ...
- 33
1
vote
4
answers
407
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Rotation systems. Problem interpreting an equation
In this equation:
$$
\mathbf a_i\overset{\rm def}{=}\left(\frac{d^2\mathbf r}{dt^2}\right)_i=\left(\frac{d\mathbf v}{dt}\right)_i=\left[\left(\frac{d}{dt}\right)_r+\boldsymbol\Omega\times\right]\left[...
- 1,629
1
vote
1
answer
246
views
Four-vector differentiation (E-M Euler-Lagrange eq.)
$$\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
1
vote
2
answers
159
views
One object moves along the cycloid at a constant rate, how about its acceleration? [closed]
We know that the parametric equation:
$$x=R(\theta+\sin(\theta))$$
$$y=-R(1+\cos(\theta))$$
and the constant velocity $c$.
How do I prove that the acceleration of the object in the $y$ direction is ...
- 21
1
vote
2
answers
428
views
Deriving the rotational Lagrangian dynamics of a quad copter
I am reading a paper deriving the dynamics of quadcopters. There are no cross-terms in the dynamics of a quadcopter so the translational and rotational energies can be separated. The problem is the ...
- 13
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
1
vote
2
answers
111
views
Derivatives of polar coordiantes?
I'm a undergraduate and I was reading about the polar coordinate system specifically this paper. I don't understand the term: $$\frac{de_r}{d\theta} = e_\theta \text{, and } \frac{de_\theta}{d\theta}...
- 13
1
vote
2
answers
167
views
Differentiate the Lagrangian wrt. momentum?
Given
$$
L=L(t, x_i, \dot x_i)
$$
as a function of generalized coordinates/velocity, and
$$
p_i:=\frac{\partial L}{\partial \dot x_i},
$$
how can we calculate
$$\frac{\partial L}{\partial p_i}?$$
- 53
1
vote
3
answers
427
views
Divergence, gradient and differentiation - radial irrotational fluid flow
Given a fluid with the steady spherically symmetric flow with only radial velocity $\vec v(r)$.
We need to evaluate $ \vec v \cdot \nabla \vec v $.
From vector calculus $$ \vec v \cdot \nabla( \vec ...
- 179
1
vote
1
answer
2k
views
What is the time derivative $\frac{d}{dt}(\exp(\hat A))$ of operator exponential $\exp(\hat{A})$? [duplicate]
Assuming that the operator $\hat{A}$ does not necessarily commute with $d \hat A/dt$, what is $$\frac{d}{dt}(\exp(\hat A))~?\tag{1}$$ Is it $\exp(\hat A(t)) \frac{d \hat A}{dt}$ or $\frac {d\hat A}{dt}...
- 69