All Questions
Tagged with differentiation homework-and-exercises
290 questions
0
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3
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82
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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
0
votes
1
answer
61
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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
$$ \frac{d}{dE} \left[ \theta(E-H(x,p;V)) ...
- 2,180
0
votes
1
answer
155
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Finding the Euler-Lagrange equation for a scalar field
Consider a scalar field with the following lagrangian density:
$$\mathscr{L}=-\frac{1}{2} \partial_{\mu} \phi \partial^{\mu} \phi-V(\phi).$$
I want to find the corresponding Euler-Lagrange equation, ...
- 69
0
votes
1
answer
60
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How can I prove this relation between derivatives? [closed]
Consider coaixialcable with TEM. Nonstatic fields are being considered, i.e situation obeys $\nabla \times \mathbf {E}=-\frac{\partial \mathbf{B} }{\partial t} $
If I let a eletric field be described ...
- 1
0
votes
1
answer
136
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Covariant derivative
$f(T,B)$ where $T = T(t)$, $B = B(t)$ and $$f_B(T,B) = \frac{\partial}{\partial B} f(T,B)$$
Now
\begin{eqnarray} \nabla^\alpha \nabla_\gamma f_B &=& g^{\alpha \sigma} \nabla_\sigma \nabla_\...
- 1
0
votes
2
answers
96
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Derive a new equation from $m=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}$ [closed]
$$m=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}$$
$$\implies m^2=\frac{m_0^2}{1-\frac{v^2}{c^2}}$$
$$\implies m^2c^2-m^2v^2=m_0^2c^2$$
Differentiating the equation,
$$2m \;dm\;c^2-2m\;dm\;v^2-2v\;dv\;m_0^2=0$...
user286470
0
votes
1
answer
159
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Relationship between derivatives of tensors in different Cartesian coordinate systems
I'm new to tensor calculus: I'm reading a little introductory book whose title is "Quick Introduction to Tensor Analysis", written by R.A Sharipov. I've reached the section called ...
- 540
0
votes
2
answers
139
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How do I find the unit vector for the tangent? [closed]
A body moves in the track from A to B to C and so on. The magnitude of the velocity in any given moment is
$|\vec V|$=$t^3$$[m/s]$. The body arrives to point B at $t$$=$$2$$[s]$. The line in the draw ...
- 3
0
votes
1
answer
27
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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:
$$
u = ...
- 67
0
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1
answer
197
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Derivation of Thermodynamic Relationships
I am currently working through some problems in thermodynamics. I am given the following relationship:
$$TdS = C_vdT + l_v dV = C_pdT + l_p dP$$
where $l_p$ and $l_v$ are some functions of state ...
- 25
0
votes
1
answer
31
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Can someone please explain me how this came? [closed]
I am not getting how above equation is derived using cylindrical coordinates transformations.
This is from page 36, Mathew Sadiku
- 27
0
votes
1
answer
84
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Identity of covariant derivative
I was reading about Einstein-Hilbert action, and in some point in this page they use this identity
$$\sqrt{-g}A^{a}_{;a}=(\sqrt{-g}A^{a})_{,a}$$
I Know that $\nabla_{\sigma}g_{\mu\nu}=0$. And $g$ ...
- 439
0
votes
2
answers
1k
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Prove the Total Mechanical Energy of the System is Conserved via Differential Equations [closed]
Consider the dynamics of a particle P shown: Particle in 3D space with Radius r
Newton's second law states that:
$$\frac{d}{dt}(m\dot r) = \mathbf F$$
where, $\boldsymbol{r}$ is the position vector ...
- 129
0
votes
1
answer
595
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How to efficiently compute the commutator $[\hat{r},\nabla^2]$?
Given a system with Hamiltonian $ \hat{H} = \frac {\hat{p} ^2}{2m} + \hat{V}(r)$ in a certain state $|\psi \rangle$, I want to find if $\langle r \rangle$ varies with time.
From
$$ i \hbar\frac {d ...
- 351
0
votes
1
answer
99
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Van der Waals cycle calculation [closed]
For the Van der Waals gas we get a cycle consisting of 2 isobaric and
2 isenthalpic processes.
We are given $T_1$,$T_3$ and $v_1$,$v_3$. And we want to calculate the
efficiency.
Attempt of ...
0
votes
1
answer
280
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How can one evaluate the expression: $\nabla_{i}\nabla_{j}\left(\frac{1}{r}\right)$, such that $i,j = x,y,z$?
I'm familiar with the Laplacian, but I'm unsure how to evaluate
$\nabla_{i}\nabla_{j}\left(\frac{1}{r}\right)$, such that $i,j = x,y,z$,
with this notation.
This is my attempt, assuming
$r=\left(...
- 32
0
votes
2
answers
310
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Canonical momentum of a 4-vector field
In a four-vector field theory,
we have a given Lagrangian:
$$\mathscr{L} = C_{1} (\partial_{\nu} A_{\mu}) (\partial^{\nu} A^{\mu}) + C_2 (\partial_{\nu} A_{\mu}) (\partial^{\mu} A^{\nu}) + C_3 A_{\mu} ...
- 112
0
votes
1
answer
349
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Derivative of the energy functional [closed]
I am trying calculate the functional derivative of the expectation value of the energy,
$$
E=\frac{\sum_{p,q}C_{p}^{*}C_{q}H_{pq}}{\sum_{p,q}C_{p}^{*}C_{q}\delta_{pq}}
.$$
With respect to $C_{p}$, ...
- 105
0
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1
answer
482
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Partial Derivative of a scalar (absolute distance) with respect to its position vector
Imagine we want to take the partial derivative of a quantity, we will call it $\rho_i = f(F(r_{ij}))$ with respect to a particle's position vector, $\vec{r}_k$.
In mathematical terms, this would be ...
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0
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1
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213
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Lagrange's equation derivation Kinetic energy
I'm trying to reach Lagrange's equations by D'alembert's principle.
$$\sum_{i=1}^N (m_i\ddot{\mathbf{x}}_i - \mathbf{F}_i)·\frac{\partial\ddot{\mathbf{x}}_i}{\partial q^\alpha}=0$$
or
$$\sum_{i=1}^N ...
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0
votes
2
answers
4k
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Why is covariant derivative a tensor?
I am trying to prove that the covariant derivative is a tensor (ie it transforms well under a change of coordinates) but I can't succeed to it.
Here is the definition of the covariant derivative :
$$...
- 1,560
0
votes
2
answers
388
views
Covariant derivatives of null tetrads
I am trying to understand the Newman Penrose null tetrads and facing some problems. Given $\ell_k$ is a null tetrad in Newman-Penrose formalism, what is $\ell_{k;i}=?$
0
votes
1
answer
104
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Snowball change in radius
So in this question I do not understand what derivative to use and how to relate it to the problem.
A spherical snowball with an outer layer of ice melts so that the volume of the snowball decreases ...
- 1
0
votes
2
answers
9k
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What is the derivative of an angle? [closed]
What is the derivative of an angle? I don't understand
0
votes
1
answer
506
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Divergence on tensor product [closed]
Can someone explain how first equation can be expanded as third equation?
I'm familiar with vector calculus, but not so familiar with tensor calculus, though I know all the definitions.
I don't have ...
- 11
0
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1
answer
2k
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What is the curl of $k\hat{r}/r^n$?
I'm trying to find the curl of ${\bf F}(r) = k \hat{r}/r^n$. I think that this converts to:
$$
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
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
0
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1
answer
72
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Calculating motion of equation in tensor form
for the Lagrangian density $$\mathscr{L}=\frac{1}{2}(\partial_{\mu}A^{\mu})^2$$
how can I get this $$\frac{\partial{\mathscr{L}}}{\partial(\partial_{\mu}A_\nu)}=(\partial_\rho A^\rho)\eta^{\mu\nu}$$
...
- 337
0
votes
1
answer
756
views
Variation of a tensor
Let a change of coordinates be given by $x^{\mu}\to x^{\mu '}=x^{\mu}+\varepsilon \xi^{\mu}(x)$ with epsilon a small quantity.
Given a tensor $T$ we define $\delta T:=T'(x)-T(x)$. I guess this means $...
- 221
0
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1
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126
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Clarification about some steps in the derivation of the Lie derivative (mechanics)
First of all, this question may seem to be undefined, because I'm not sure how to connect this (to me) newly introduced concept with the abstract notion of the Lie derivative. I'm not even sure if I ...
- 912
0
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1
answer
106
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I need help with divergence and gradient? [closed]
$$A_z = \mu{\frac{e^{-jBr}}{4\pi r}}∫I(z')e^{jBz'\cos\theta}dz'$$
Midway into my question, I want to compute:
$$-j\left( \frac{\nabla(\nabla\cdot A) }{w\mu\varepsilon} \right).$$
Symbols like $ w, \...
- 113
0
votes
1
answer
68
views
The vector r points from $P'(x',y',z')$ to $P(x,y,z)$ [closed]
For some reason this question is giving me a hard time :(
The vector $r$ points from $P'(x',y',z')$ to $P(x,y,z)$.
(a) Show that if $P$ is fixed and $P'$ is allowed to move, then $\nabla'(\frac{1}{r}...
- 817
0
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1
answer
34
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Why does the power in an inductor equal what it does?
I understand that power is that rate at which work is done and that because of this the power in an inductor is equal to
$$P=\frac{d}{dt} \left(\frac12Li^2\right).$$
I also understand that the power ...
0
votes
1
answer
54
views
Differentiate wave speed, don't understand
The speed $v$ of some wave is $ω/k$ and I want to differentiate this with respect to $k$. Apparently this equals:
$dv/dk = d(ω/k)/dk-ω/k^2$
But I don't understand why. Isn't this just saying "the ...
- 33
0
votes
1
answer
154
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Finding the Lagrangian from the derivative of position
I have to find the Lagrangian for a system.
In the point of interest I have come up with the following position coordinates:
$$x = Rcos(\omega t)+\ell sin(\phi)$$
and
$$y = Rsin(\omega t)-\ell cos(\...
- 2,577
0
votes
1
answer
69
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D'Alembert Solution to 1+1D wave equation - integration step
I am working through d'Alembert's solution to the 1+1D wave equation using the substitution of canonical coordinates. I have an initial condition of: $$u_{t}(x,0) = g(x) $$ with a general solution ...
0
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0
answers
72
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A calculation question about Taylor expansion in Altland and Simons p 106, the gutter like potential
I have a question regarding the book condensed matter field theory by Altland Simons p 106. In their a gutter like potential is given and it is required to calculate the fluctuation $\delta V_{tension}...
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0
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0
answers
36
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How to evaluate a non-banal derivate?
I need to evaluate the following derivate:
$$\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]$$
where $\Psi$ is a ...
0
votes
1
answer
81
views
Volume of a two-dimensional sphere in a fixed three sphere geometry
I'm just starting to read Hartle's Gravity and he gives the following equation for the volume of a 2D sphere of radius $r$ if space was a fixed 3-sphere geometry on page 20.
$$V = 4\pi a^3\left(\frac{...
- 5
0
votes
2
answers
194
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Variation of the contravariant component of the metric respect to the covariant component of the metric
I am recently studying general relativity and it is a bit difficult for me to handle the rise and fall of indices in some calculations. My specific question is how could I find this variation?
$$\frac{...
0
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0
answers
108
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What does $\nabla'$ mean? [duplicate]
In D.J Griffiths Electrodynamics (Page 173) it says, $\nabla' |\vec{x}| = \frac{\hat{x}}{x^2}$. However by my calculation $\nabla |\vec{x}| = -\frac{\hat{x}}{x^2}$ so what does the $\nabla'$ signify?
0
votes
2
answers
145
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Spin coherent state path integral derivation
I'm trying to follow the exposition of spin coherent state path integral presented in Condensed Matter Field Theory by Altland and Simons (section 3.3, Page 134-142), and I have a problem with the ...
- 3
0
votes
1
answer
399
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Killing equation in coordinates
In proving that it is possible to write the killing equation in coordinates as $$L_X g=0\iff X_{\alpha;\beta}+X_{\beta;\alpha}=0$$
I have read that the key observation, to write the equation in ...
- 187
0
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1
answer
1k
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Commutator of covariant derivative and field $F_{\mu \nu}$
I am working with the covariant derivative and trying to show that the commutator of this derivative
$[D_\mu , D_\nu]$ is proportional to the field $F_{\mu \nu}$. That is, I need the final term to
be ...
user276724
0
votes
2
answers
89
views
Independence of kinetic energy with respect to $\rho$ when potential energy is independent of $\phi$
Reference material (free access right now): Shankar
On page 81 of Shankar, (paraphrasing) he states that:
$ 1: T$, the kinetic energy in polar coordinates, depends solely on $\frac{d}{dt} \phi$ if $\...
- 182
0
votes
1
answer
347
views
Projectile motion - differentiating the equation of trajectory to find the maximum height
I have the equation of trajectory:
$ y = x\tan \theta - {\displaystyle gx^2 \over \displaystyle2u^2\cos^2 \theta}$
I also know that the maximum height is given by:
${\displaystyle u^2 \over\...
- 13
0
votes
1
answer
65
views
$R$-Symmetry of gauge field
Suppose $V$ is a superfield scalar under R-transformations. This means that under an R-transformation $V\mapsto V'$ where $V'(x,\theta,\bar{\theta})=V(x,e^{-iK}\theta,e^{iK}\bar{\theta})$. What is ...
- 3,985
0
votes
1
answer
34
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Help with deriving simple heat equation [closed]
$$j^{q}=\frac{1}{2} n v[\varepsilon(T[x-v \tau])-\varepsilon(T[x+v \tau])]$$
To this:
$$j^{q}=n v^{2} \tau \frac{d \varepsilon}{d T}\left(-\frac{d T}{d x}\right)$$
At first I was thinking of using ...
- 13
0
votes
2
answers
84
views
Acceleration in a non-inertial reference frome - derevation
The general velocity equation for a point B in on body rotating and translating about point A with respect to the inertial reference frame say 'xyzo' can be expressed as,
$\vec{r_{B/o}} = \vec{r_{A/o}...
- 17
0
votes
0
answers
63
views
Explicit form for $\frac{\delta u^\alpha}{\delta g^{\mu\nu}}$?
The title says it all: Is there a closed form expression for the following variational derivative,
$$
\frac{\delta u^\alpha}{\delta g^{\mu\nu}},
$$
where $u^\alpha$ is the four-velocity of a massive ...
- 444