All Questions
Tagged with differentiation differentiation or
1,900 questions
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Laplacian and Dirac Delta function
Although I find it mathematically dubious, we said that $$\Delta \frac{1}{r} ~=~ -4\pi \delta^3({\bf r}).$$ Now, I was wondering is there a similar relation to the delta function if we look at $$\...
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2
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Trouble with derivation in an equation for Newton's Law of Angular Motion
I'm an autodidact and can't follow the part after "it is easily seen that"... which is the 31st equation:
Shouldn't it be:
$m_i\,{\bf r}_i\times \frac{d^2{\bf r}_i }{dt^2}= \frac{d}{dt}(m_i r_i \...
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1
answer
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Time derivative of the state vector as expressed in abstract Hilbert space vs. as a wavefunction
The Schrodinger equation in Hilbert space is expressed as : $$\frac{\partial}{\partial t} \psi(t) = \frac{-i}{\hbar}H\psi(t). $$
Here $\frac{\partial}{\partial t} \psi(t) \equiv \psi'(t) \equiv\lim \...
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1
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Taylor series expansion of $\ln$ and $\cosh$ in distance fallen in time $t$ equation
I want to find the Taylor expansion of $y=\frac {V_t^2}{g} \ln(\cosh(\frac{gt}{V_t}))$
I have tried using the fact $\cosh x= \frac {e^x}{2}$ for large t, which works, I just need help on small values ...
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1
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What is the infinitesimal work done when the force is given by the gradient of a scalar function that depends both on position AND time?
The title is slightly confusing but I didn't know how else to phrase my question.
Basically, this is the situation:
When the force applied to a particle is given by the gradient of a scalar function ...
- 1,373
3
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1
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133
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Contradiction of a scalar product
Can anyone resolve this contradiction:
$$\vec{r}\cdot\dot{\vec{r}}=\frac{1}{2}\frac{d}{dt}\left(\vec{r}^2\right)=\frac{1}{2}\frac{d}{dt}\left(\left|\vec{r}\right|^2\right)\equiv\frac{1}{2}\frac{d}{dt}...
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What is the difference between $\nabla _{\sigma} $ and $ \nabla^{\sigma}$?
What is the difference between:
$\nabla _{\sigma} $ and $ \nabla^{\sigma}$?
I've been told that the first is the covariant derivative, however I'm just starting a course on spacetime geometry and ...
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1
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Are covariant derivatives of Killing vector fields symmetric?
I'm reading the Lecture Notes on General Relativity by Matthias Blau, and in section 9.1 (point 1) he writes:
Let $K^\mu$ be a Killing vector field, and ${x^\mu(\tau)}$ be a geodesic. Then the ...
user24999
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1
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Index Notation with Del Operators
I'm having trouble with some concepts of Index Notation. (Einstein notation)
If I take the divergence of curl of a vector, $\nabla \cdot (\nabla \times \vec V)$ first I do the parenthesis:
$\...
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4
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Conserved quantities and total derivatives?
I am having a bit of a crisis in understanding of the physical meanings of total derivatives.
When a quantity $\rho$ (be it a vector or a scalar) is said to be conserved, then (mathematically) $$\...
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1
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Question concerning the Feynman Lectures of Physics
I am reading the Feynman lectures and at this point http://www.feynmanlectures.caltech.edu/I_13.html#Ch13-S3 it says as follows:
The time derivate of the potential energy is
$\begin{equation}
\...
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1
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What is meant by incompressible flow?
What is meant by incompressible flow?
The density of the fluid is a constant, $\rho = constant$
The density of a fluid has a spatial dependence but remains constant in time, $\rho = \rho(\mathbf{r})$
...
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3
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Ordering of differential operators
If we write something like:
$\partial_a X_{\mu} \partial^a X^{\mu}$
Does that mean the first derivative is only applied to the first X?
($\partial_a X_{\mu})( \partial^a X^{\mu}$)
Or is the ...
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1
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How to derive the Schwarzschild metric?
I'm having trouble differentiating the following when making a change of co-ordinates to determine the Schwarzschild metric.
$$r'^{2}=r^{2}C(r)$$
Then taking the total derivative of both sides, the ...
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Fermion propagator is not a Grassmann-odd object?
Is the following differentiation correct:
$$ \frac{\delta}{\delta\eta\left(z\right)}\int d^{4}yS_{F}\left(z-y\right)\eta\left(y\right) = S_F\left(z-z\right)$$
where $\eta$ is a Grassmann-valued ...
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Name this Mulltivariable Calculus Theorem
In Robert Wald's book General Relativity a multivariable calculus theorem is cited on page 16, which states:
If $F:\mathbb{R}^n\mapsto \mathbb{R}$ is $C^{\infty}$ then for each $a=(a^1,...,a^n) \in ...
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1
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Covariant derivative of a vanishing tensor component [closed]
Is the covariant derivative of a vanishing tensor component necessarily zero?
38
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5
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Equivalence between Hamiltonian and Lagrangian Mechanics
I'm reading a proof about Lagrangian => Hamiltonian and one part of it just doesn't make sense to me.
The Lagrangian is written $L(q, \dot q, t)$, and is convex in $\dot q$, and then the ...
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Differentiate wave speed, don't understand
The speed $v$ of some wave is $ω/k$ and I want to differentiate this with respect to $k$. Apparently this equals:
$dv/dk = d(ω/k)/dk-ω/k^2$
But I don't understand why. Isn't this just saying "the ...
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1
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Covariant derivative as a tensor
$$\nabla_{j} v^{i}~=~g^{ik}\nabla_{j}v_{k}.$$
Does this equality involve an intermediate step, where I take the metric inside the derivative, and then use the fact that covariant derivative of the ...
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Help deriving the general linear wave equation $d^2y/dx^2=(1/v^2)d^2y/dt^2$ [closed]
How do I derive the General Linear Wave Equation $$d^2y/dx^2=(1/v^2)d^2y/dt^2?$$
My teacher differentiated the general wave function $f(x + vt)+g(x - vt)$ twice with respect to both variables to get ...
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2
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460
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Taking time derivative of two dependant variables
I'm not entirely sure if this is correct. I have to take the time derivative of the following:
$$\frac{d}{dt}mr^{2}\dot{\phi}$$
Now, both $r$ and $\dot{\phi}$ depends on the time $t$, so I have to ...
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226
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How to do this index notation differentiation?
I am studying classical Maxwell fields and I am stuck on this differentiating part. How can I derive the result given below ?
$$\dfrac{\partial}{\partial(\partial A_{\mu}/\partial x_{\nu})} \left(2\...
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4
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What is the current of a capacitor when the derivative of voltage is undefined?
This is from the textbook I am reading:
I know this equation for capacitors:
$$i=C\cdot \frac { dv }{ dt }$$
Here is my question: how can diagram (a) be allowed if the derivative of the voltage ...
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D'Alembertian for a scalar field
I have read that the D'Alembertian for a scalar field is
$$
\Box = g^{\nu\mu}\nabla_\nu\nabla_\mu = \frac{1}{\sqrt{-g}}\partial_\mu (\sqrt{-g}\partial^\mu).
$$
Exactly when is this correct? Only for $...
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About Divergence in polar coordinates
I've got a conductor in a cylinder shape that is rotating with angular velocity $\omega$ around its axis, that correspond to the $z$ axis
I want to calculate the electric field and the density of ...
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How is the direction of the instantaneous acceleration determined?
I know from the text book that the direction of velocity at any point on the 2D path of an object is tangential to the path at that point and is in the direction of motion. But how would one determine ...
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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
$$y = Rsin(\omega t)-\ell cos(\...
- 2,577
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Detail of deriving Berry Curvature From Berry Connection
The Berry curvature of the $n^{\mathrm{th}}$ eigenstate of Hamiltonian $H$ for the vector of external parameters $\vec{R}$ can be derived in part by writing the following two lines:
$$
B^n(\vec{R}) \...
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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
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3
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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? ...
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Physical motivation for differentiation under the integral
I am thinking about the mathematical process of "differentiating underneath the integral", i.e. applying the theorem $$\partial_s \int_{-\infty}^\infty f(x,s)\,dx=\int_{-\infty}^\infty \partial_s f(x,...
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Lagrangian Mechanics - Commutativity Rule $\frac{d}{dt}\delta q=\delta \frac{dq}{dt} $
I am reading about Lagrangian mechanics.
At some point the difference between the temporal derivative of a variation and variation of the temporal derivative is discussed.
The fact that the two are ...
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1
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What is a covariant derivative in gauge theory?
I've been studying electroweak theory and you need to keep the Lagrangian covariant by introducing covariant derivatives. What is a covariant derivative? And what does it mean to keep the Lagrangian ...
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Can You Obtain New Physics from the use of Fractional Derivatives?
I was curious if anyone could give me an example of the use of fractional derivatives in physics and explain what they offer that "conventional" mathematics does not (in terms of new physics and not ...
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3
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Derivative with respect to a vector is a gradient?
I've encountered in some books (and even completed an exercise from the Goldstein by using it), a strange notation that seems to work exactly like a gradient, I have tried to look for an explanation ...
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1
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381
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Higgs mechanism in QED
I'm trying to understand the Higgs mechanics. For that matter, I'm exploring the possibility of giving mass to the photon in a gauge-invariant way. So, if we introduce a complex scalar field:
$$ \phi=...
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1
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Neglecting second order differentials
I am currently doing some Lorentz invariance exercises considering infinitesimal Lorentz transformations, and have been told to neglect second order differentials.
It's not the first time I have come ...
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When does the total time derivative of the Hamiltonian equal its partial time derivative?
When does the total time derivative of the Hamiltonian equal the partial time derivative of the Hamiltonian? In symbols, when does $\frac{dH}{dt} = \frac{\partial H}{\partial t}$ hold?
In Thornton &...
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When motion begins, do objects go through an infinite number of position derivatives?
This might be a very vague and unclear question, but let me explain. When an object at rest moves, or moves from point $A$ to point $B$, we know the object must have had some velocity (1st derivative ...
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Derive vector gradient in spherical coordinates from first principles
Trying to understand where the $\frac{1}{r sin(\theta)}$ and $1/r$ bits come in the definition of gradient.
I've derived the spherical unit vectors but now I don't understand how to transform ...
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Scale-invariant differential operator
For example, the differential operator Laplacian is
$$\nabla^2 = \frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}.$$
My questions are:
Is it scale-invariant?
what is scale-...
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Feynman's subscript notation
Consider this vector calculus identity:
$$
\mathbf{A} \times \left( \nabla \times \mathbf{B} \right) = \nabla_\mathbf{B} \left( \mathbf{A \cdot B} \right) - \left( \mathbf{A} \cdot \nabla \right) \...
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How would one show that a nonabelian field strength tensor transforms in a certain way under a local gauge transformation?
How would one show that the nonabelian ${F_{\mu\nu}}$ field strength tensor transforms as $${F_{\mu\nu}\to F_{\mu\nu}^{\prime}=UF_{\mu\nu}U^{-1}}$$ under a local gauge transformation? Rather than ...
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Gravitational force exerted by a rod on a point mass
I have doubts with the solution of a certain problem. I will give the entire solution below and will lay out my doubts as well.
A point mass $m_1$ is separated by a distance $r$ from a long rod of ...
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3
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Why and how maximum force is $\frac{dF}{dx}=0$? [closed]
In an certain question my teacher asked to find the maximum force. She said that the maximum force in electrostatics means $\frac{dF}{dx}=0$. Why is it like that?
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Meaning of "Gradient with respect to coordinates of particle" in SPH
I'm currently trying to implement a simple SPH simulation based on a variety of papers. However as I'm not a trained physicist nor mathematician I have a small issue with the following notation and ...
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When we take time derivative of a function of time, then is the result another function of time, again?
(I'll try to explain my question by one known example), for example where the velocity is a function of time v(t) then its time derivative (which is acceleration: $a=\frac {dv}{dt}$) is another ...
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1
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In Newtonian pressure, what type of function is force?
This is pressure in Newtonian mechanics:
$$P=\frac {dF}{dA}.$$
What does this mean?
(Doesn't it mean that force is a function of area?)
What type of function is force?
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-8
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3
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Is there any other mathematical tool to measure velocity, instead useing derivative? [closed]
To measure velocity we use derivative
$$v=\frac {dr}{dt}.$$
Is the any other mathematical tool to do this?.
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