All Questions
Tagged with differentiation homework-and-exercises
290 questions
0
votes
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
1
vote
1
answer
483
views
Tensors and derivatives
I am a maths student taking a module in (the mathematics of) Relativity so I get quite confused when looking for stuff that may help me understand where I go wrong in certain questions as I'm not ...
- 11
1
vote
1
answer
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
105
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How to derive $\frac{d}{d\lambda}=-\mathbb{y}\cdot\nabla$ for $\frac{1}{\lvert\mathbb{x}-\lambda\mathbb{y}\rvert}$?
When talking about the multipole expansion of an electromagnetic potential, my professor noted that for the function
$$\tag{1}\frac{1}{\lvert\mathbb{x}-\lambda\mathbb{y}\rvert},$$
the two operators
...
- 215
3
votes
4
answers
902
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Multiplicity Identity in Kittel's Thermal Physics
On page 25 of Kittel's Thermal Physics, the author derives the multiplicity of $N$ harmonic oscillators with total quanta of energy $n$, $g(N,n)$.
He writes
\begin{align}
g(N,n) &= \lim_{t\...
- 129
1
vote
1
answer
345
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Starting from an expression of E(V) and P(V) for the Birch-Murnaghan's equation of state, is there a way of obtaining an expression for E(P)?
I have a function, an equation of state named the Birch-Murnaghan's equation of state, in which the Energy ( $E$ ) is a function of the Volume ($V$), where $E_{0}$, $V_{0}$, $B_{0}$ and $B_{0}^{'}$ ...
- 131
5
votes
3
answers
1k
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Vlasov equation, Maxwell distribution
I have the Maxwellian distribution:
$$f(v)=n\left(\frac{m}{2\pi kT}\right)^{\frac{3}{2}}\exp\left(-\frac{mv^2}{2kT}\right)$$
I have to show that it is a solution to the Vlasov equation:
$$\frac{\...
-1
votes
1
answer
569
views
Confusion in differentiation in physics problem [closed]
Here, we had to find theta such that the denominator has the maximum value. Being new to differentiation I basically didn't understand how differentiation solved the purpose:
I basically didnt ...
- 997
3
votes
2
answers
2k
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For the Yang-Mills field strength defined as a commutator, why does the $A_\nu\partial_\mu - A_\mu\partial_\nu$ term vanish?
In basically every QFT book the Yang-Mills strength tensor $F_{\mu\nu}$ is defined as $$F_{\mu\nu}=[D_\mu,D_\nu]$$
where $D_\mu$ is the covariant derivative $$D_\mu=\partial_\mu-A_\mu$$ and $A_\mu$ is ...
- 1,674
1
vote
1
answer
216
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Second covariant derivative, computation problem
I am having a question on the wikipedia article http://en.wikipedia.org/wiki/Second_covariant_derivative
Using the notation therein I don't get why
$(\nabla_{u}\nabla_{v}w )^a=u^c\nabla_{c}v^b\nabla_{...
- 123
6
votes
5
answers
8k
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Covariant Derivative of Kronecker Delta
I am reading Carroll's book on GR right now, and I ran into a little trouble in his chapter 3 on curvature. He is establishing the properties of the covariant derivative, and claims that the fact that ...
- 147
0
votes
1
answer
45
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Equations of motion acceleration doubt
So i was going through some text today morning. Where it said
$$ a = \frac{vdv}{dx} $$
So they then went on to,
$$ vdv = adx \\ \implies \int vdv = \int adx$$
But,I am very certain acceleration is ...
- 23
-1
votes
1
answer
22k
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Maximum electric field of a circular ring
How do you differentiate the equation for electric field of uniform ring
$$ E_x = \frac{kxQ}{(x^2+r^2)^{3/2}} $$to get the maximum at a point? My book says $x = r/\sqrt2$. I tried differentiating ...
- 307
7
votes
2
answers
3k
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Covariant derivative of a covariant derivative
I'm trying to find the covariant derivative of a covariant derivative, i.e. $\nabla_a (\nabla_b V_c)$.
This is something I've taken for granted a lot in calculations, namely I though that by the ...
- 623
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1
answer
58
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Differential Operator
I am trying to understand the following expression
\begin{eqnarray}
e^{-ik.x}D_{\mu}D^{\mu}e^{ik.x} & = & e^{-ik.x}(i\partial_{\mu}+A_{\mu})(i\partial^{\mu}+A^{\mu})e^{ik.x}\\
& = & e^{...
- 53
0
votes
2
answers
148
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Calculate divergence via partial derivative [closed]
I need to calculate the divergence and curl for a vectorfield. I've done that before so that's no problem :) Or I've done it using partial derivative, maybe there are multiple ways to solve for ...
- 35
1
vote
1
answer
288
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Help with relativistic notation (Derivative of Lagrangian)
I am trying to learn QFT, but I haven't taken a course in general relativity so the relativistic notation stuff is taking me a bit to get used to. I do not understand how to do the following.
For a ...
- 1,001
-1
votes
1
answer
762
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Uncertainty in Range of Projectile [closed]
If we are given that a projectile is launched with velocity 10m/s at an angle of $45^\circ$ and uncertainty in angle is of $0.5^\circ$ . What is the uncertainty in the range of projectile.
The problem ...
- 53
0
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1
answer
72
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Calculating motion of equation in tensor form
for the Lagrangian density $$\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}$$
...
- 337
2
votes
1
answer
2k
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Divergence of inverse of metric tensor
I know that the Levi-civita connection preserves the metric tensor. Is the divergence of the inverse of metric tensor zero, too?!
I'm not so familiar with the divergence of the second ranked tensor. ...
- 119
2
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0
answers
345
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Supercovariant Derivative action
My query is with Weinberg Vol3 equation just above 26.7.22
Weinberg follows Majorana Superfield formalism. Where, covariant derivative is defined as,
$$D_{R\alpha}=-\epsilon_{\alpha \beta}\frac{\...
- 142
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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
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 ...
- 198
0
votes
1
answer
91
views
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:
$$(...
- 21
-2
votes
1
answer
286
views
To prove, $\nabla.(\nabla\phi \times \nabla\psi)$ =0 [closed]
Please Help me solving the problem using levi-cevita symbol :
Prove That, $\nabla.(\nabla\phi \times \nabla\psi)$ =0 where $\phi =\phi(x,y,z)$ & $\psi=\psi(x,y,z)$
- 300
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1
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126
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Clarification about some steps in the derivation of the Lie derivative (mechanics)
First of all, this question may seem to be undefined, because I'm not sure how to connect this (to me) newly introduced concept with the abstract notion of the Lie derivative. I'm not even sure if I ...
- 912
1
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1
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70
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Derive an equation related to magnetism [closed]
Solve the equations for $v_x$ and $v_y$ :
$$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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0
answers
28
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Evaluating derivatives with respect to certain vector axis
So, I am trying to work in Spherical coordinates. I have a predefined fixed axis, $\hat{v}_0$, so that $\alpha=\vec{r}.\hat{v}_0$ Now, I am interested in the following:
\begin{equation}
f(r,\alpha)=\...
- 490
3
votes
1
answer
271
views
How is $ \frac{dv}{ dt} = a $?
I know how , in the physical sense -
$$\frac {dv}{dt} = a$$
But, even after thinking a lot, I am not able to see the fault in this -
$$\frac {dv}{dt} = \frac {d(st^{-1})}{dt}
= \frac {sd(t^{-1})}...
- 141
2
votes
1
answer
2k
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Derivatives with upper and lower indices
I'm studying classical and quantum field theory, but evaluating derivatives of fields (scalar and/or vector) described with upper and lower indices is somewhat new to me. I'm trying to evaluate
$$\...
- 187
0
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2
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17k
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What is the derivative of $\dot{\theta^2}$? [closed]
$$\frac{d}{dt}(\dot{\theta^2}) =? 2\dot{\theta}\ddot{\theta}$$
is this correct, or am I missing something?
- 53
-1
votes
1
answer
518
views
Forced damped harmonic motion, angular frequency at which amplitude is maximum. differentiation [closed]
$$A_0 = \frac{(F_0/m)}{\sqrt{(\omega_0^2-\omega_d^2)^2+b^2\omega_d^2/m^2}}$$
How would I differentiate this with respect to the driven angular frequency (equating to zero) in order to obtain the max ...
- 1
6
votes
2
answers
12k
views
Variation of square root of determinant of metric, $\delta g$ [closed]
I am trying to calculate
$$ \frac{\partial \sqrt{- g}}{\partial g^{\mu \nu}},$$
where $g = \text{det} g_{\mu \nu}$.
We have
$$ \frac{\partial \sqrt{- g}}{\partial g^{\mu \nu}} = - \frac{1}{2 \...
- 329
-2
votes
3
answers
29k
views
Derivative of kinetic energy [closed]
I read that the derivative of kinetic energy=$F\cdot v$. I tried to differentiate (1/2) mv^2 with respect to time but each time I am getting $m*v$ and not $m*a*v$ which solves to $F*v$.
My efforts are ...
- 274
0
votes
1
answer
123
views
How do I set when the object isn't moving
I started studying instantaneous velocity derivatives using only now.
It may seem stupid but really I'm not sure whether that's right: I have an equation:
$$x (t) = 1.5t - 9,75t³$$
To set the time ...
- 107
1
vote
1
answer
1k
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Covariant derivative ordering
I was working on a problem involving Bianchi identities, in a particular case I have to take the covariant derivative of the following, which indeed is the Ricci tensor in linearised limit
$$r^{\mu}_{\...
- 278
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
0
votes
1
answer
2k
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Classical Mechanics, The Theoretical Minimum: error in answer to partial derivatives exercise? [closed]
I'm reading Leonard Susskind's Classical Mechanics, The Theoretical Minimum, and I'm on the interlude on partial derivatives. There is an exercise that asks you to find all of the first and second ...
2
votes
2
answers
4k
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Total time derivative of magnetic vector potential $A$
I am looking at this document, which tries to establish the Lagrangian of the Lorentz force. Everything is fine, but I don't see why:
$$\frac{dA_i}{dt}=\frac{\partial A_i}{\partial t}+\frac{\partial ...
- 604
1
vote
1
answer
392
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Proof for Negele and Orland equation (2.34)
The equation (2.34) of Negele and Orland has
$$\mathcal H_\text{A}(\hat{\mathbf p},\hat{\mathbf x}) =
\frac{1}{2m}\left(\hat {\mathbf p} - \frac e c \mathbf A(\hat{\mathbf x})\right)^2.\tag{2.34a}$$
...
- 11
1
vote
1
answer
488
views
Varying wrt metric [closed]
I saw people write $\frac{\partial( F^{ab} F_{ab})}{\partial g^{ef}}$ as
$\frac {\partial (g^{ca}g^{db}F_{cd}F_{ab})}{\partial g^{ef}}$ in a way that exposes the dependence on the metric. but ...
- 79
1
vote
1
answer
211
views
Lagrangian formalism (demonstration)
My question is about the multiplicity of the Lagrangian to a Physics system.
I pretend to demonstrate the following proposition:
For a system with $n$ degrees of freedom, written by the Lagrangian ...
- 1,092
0
votes
1
answer
72
views
Show $\frac{\partial T}{\partial \dot q_j} = m_i \dot r_i^T\frac{\dot r_i }{\partial \dot q_j} $ [closed]
This is a basic result in lagrangian mecanics. Let $T$ be the kinetic energy, $r_i$ be the position of the $i^{th}$ particle in the system I need to show $$\frac{\partial T}{\partial \dot q_j} = \frac{...
- 1,754
2
votes
1
answer
288
views
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
0
votes
1
answer
106
views
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
1
vote
1
answer
100
views
Indexed Gradient operator on trigonometric functions
$$\nabla_{i}\nabla_{j}\Big(\frac{\sin(kR)}{R}\Big)$$ Where $R$ is the distance between particle $i,j$. And $k$ is a constant
I took $\nabla_{i}=\frac{\partial}{\partial R_{i}}$ and $\nabla_{j}=\frac{\...
- 51
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
1
vote
2
answers
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
votes
3
answers
69
views
Vector question, differentials, Electromagnetism
I was reading this demonstration of electric potential in my book:
Let $q$ be a point charge at point $P$
The Electric field created at point $M$ by $q$ is :
$$\vec{E}(M) = \...
- 685
6
votes
2
answers
4k
views
Advection operator
How are exactly $u_j\partial_ju_i$ and $u_i\partial_j u_i$ related?
And what is their relation to ($\boldsymbol{u}\cdot\nabla)\boldsymbol{u}$ and $\boldsymbol{u}\cdot(\nabla\boldsymbol{u})$ ?
I ask ...
- 1,813