Questions tagged [differentiation]
Differentiation is the set of techniques and results from Differential Calculus, concerning the calculation of derivatives of functions or distributions.
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How to take derivative of density operator?
I was just trying to confirm to myself that the following density operator
$$\rho(t) = e^{-iHt/\hbar} \rho(0) e^{iHt/\hbar}$$
fulfills the Liouville-von Neumann equation:
$$\frac{d}{dt}\rho(t) = - \...
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What's the difference between $\nabla\cdot(\rho v)$ and $\rho(\nabla\cdot v)$ as a physical intuition?
I'm currently learning on substantial derivatives in fluid mechanics and kind of understand how partial derivatives $\frac{(\partial\rho)}{(\partial t)}$ and substantial derivatives $\frac{(D\rho)}{(...
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From where does the expression of the tangential accerelation come from?
I've seen so many times that the expression of the tangential acceleration is known to be: $$a_t=\ddot{s}$$ but from the expression of the acceleration in spherical coordinates, in the tangential ...
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Sum of two state functions is not path independent
I am trying to explain the different physical meanings of the various thermodynamic potentials (before resorting to Legendre transforms, and without making appeals to statistical mechanics) and I ...
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Square Root of an Operator
NOTE: CROSS-POSTED ALSO ON MATH EXCHANGE
I'm working with quantum mechanics and i have to solve an operator equation.
I have the two canonical variables $S$ and $P$ which are then promoted to ...
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Is the derivative of the adjoint the adjoint of the derivative? [closed]
Does $\frac{d}{dx}\langle u|= (\frac{d}{dx}|u\rangle)^\dagger$? Here $|u\rangle$ is any vector in a Hilbert space and $\dagger$ is the adjoint/conjugate-transpose.
It seems to me that this should not ...
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Derivation of covariant derivative by means of parallel transport
I've studied covariant derivative in many courses so far, but I got stuck on the definition given by the teacher notes of the exam of Topological QFT.
I think that he improperly used the name "...
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Why is the regular Taylor expansion so similar to the construction of the Hamiltonian from a Lagrangian/Legendre transform?
An example of a first order Taylor expansion of a function with two variables is given by:
$$f\left(x,y\right)=f\left(x_0,y_0\right)+\left(x-x_0\right)\frac{\partial f}{\partial x}+\left(y-y_0\right)\...
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Why does $\delta \vec{r} = \delta \vec{ \theta} \times \vec{r}$?
Hello fellow physicists,
I was trying to understand some behavior on rotating objects, specifically about the formula $\vec{v} = \vec{\omega} \times \vec{r}$.
The Book (Marion, J. B. (1965). Classical ...
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"Why is $n$ held constant when taking the time derivative in the course of the Van Kampen's system size expansion?"
I follow the notation used in "stochastic processes in physics and chemistry"(p.245) by Van Kampen.
Left hand side of master equation is
$$\frac{\partial P(n,t)}{\partial t}=\cdots.$$
We ...
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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:
...
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Reduce multiplicative noise to additive noise with singular matrices
I have a stochastic differential equation as
\begin{equation}
\dot{\textbf{X}}=\textbf{A}\textbf{X}+\alpha(t)\textbf{B}\textbf{X}-\alpha^*(t)\textbf{B}^T\textbf{X}
\end{equation}
where $T$ ...
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How do we get Momentum operator from linear momentum? [duplicate]
How do we get the momentum operator $p=-i(h/2π) ∇)$?
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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 ...
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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: ...
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Cylindrical incompressible Navier-Stokes, are the derivatives commuative?
I am currently trying to understand a paper by Eckhardt et al. (https://doi.org/10.1017/S0022112007005629). In it, a transformation is performed on the cylindrical incompressible Navier-Stokes ...
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Physical Interpretation of Flowrate Time Derivative of Incompressible Fluid in Variable Volume Vessel with Single Inlet
I am an academic researcher who studies fluid mechanics of the left ventricle (the primary chamber of the heart that actually pumps blood to the rest of your body). The majority of my work focuses on ...
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Confirming an action is invariant under a supersymmetric transformation
I am studying chapter 9 of the book Mirror Symmetry, available here. My question is relating to page 156/157 where Supersymmetry is being introduced for the first time in QFT in 0-dimensions.
We are ...
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Del operator confusion [closed]
The very first thing my textbook says is that the Del operator is defined as:
$$\vec{\nabla}=\vec{a}^i\nabla_i$$
Where $\nabla_i$ is the covariant derivative and " $\vec{a}^i$ is the curvilinear ...
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What is the intuition or the derivation of covariant derivative?
I asked this question in mathematics but the answer I got was a bit too abstract for me so I hope that my fellow physicists can give me more of an intuition or an easier explaination of my question. ...
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Looking for the geometric meaning of the curl of Killing vector fields
From Killing equation
$$\nabla_\nu \xi_\mu + \nabla_\mu \xi_\nu = 0$$
it can be shown that $\nabla_\nu \xi_\mu$ is antisymmetric.
From it we can construct an antisymmetric tensor $\mathcal{A}_{\mu\nu}$...
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What is the gauge covariant derivative?
What is the gauge covariant derivative in layman's? Does it describe the kinetic energy of the gauge fields?
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Integral of derivative of delta function gives strange answer [closed]
So I've been doing some QM and I keep coming across the following type of integrals:
$$
\int f(x) \frac{\partial}{\partial x} \delta(x-x') dx.
$$
I know that I should integrate by parts but then I ...
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Covariant derivative of metric determinant with torsion
I have some troubles taking the covariant derivative of the metric determinant with torsion. Let's suppose that we take a metric such that $\nabla_\mu g_{\nu\rho}=0$. My reasoning is the following.
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Is the rate of change of duration a valid quantity?
I was wondering that, if the duration of a recurring event varies as time goes on, what would the magnitude of this quantity be measured in? For instance, if the time for an oscillation of a weighted ...
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Lie derivative of a one-form
I am going through Nakahara's textbook on geometry and topology in physics. Intuitively, I understand the definition of a lie derivative of a vector field
$$\mathcal{L}_xY = \lim_{\epsilon_\to 0}\frac{...
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The BRST variation of the gauge fixing condition
Following Polchinski volume I, p 126 onwards, The BRST variation of fields $\phi^{i}$ is given by
$$\delta_{B} \phi_{i} = - i \epsilon c^{\alpha} {\delta}_{\alpha}\phi_{i} \; .\tag{4.2.6a}$$
My ...
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About the traditional explanation of the continuity of the first derivative of a 1D wavefunction
I would like to receive some clarifications about the traditional explanation of the continuity of the first derivative of a 1D wavefunction (E.g. see the very clear answer by @ZeroTheHero ...
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Proving entropy is an exact differential [duplicate]
I have been trying to understand something that my professor explained and I have tried everything (books, my classmates' notes, internet) but I can't understand, can you please help retrieve the ...
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What are some ways to derive $\left( \boldsymbol{E}\cdot \boldsymbol{E} \right) \nabla =\frac{1}{2}\nabla \boldsymbol{E}^2$?
For each of the two reference books the constant equations are as follows:
$$
\boldsymbol{E}\times \left( \nabla \times \boldsymbol{E} \right) =-\left( \boldsymbol{E}\cdot \nabla \right) \boldsymbol{E}...
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Cross factor for dependent terms in a differential?
How do you derive a cross factor to decouple differentials into independent differentials? For example:
$$ d(PV)= PdV+VdP $$
$$ PV=\int{PdV}+\int{VdP} $$
Obviously dP and dV are related. Do you simply ...
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Why does $\vec{r}\cdot\dot{\vec{r}}=r\dot{r}$?
Why is $$\vec{r}\cdot\dot{\vec{r}}=r\dot {r}$$ true? Before saying anything, I have seen the proofs using spherical coordinates for $$\dot{\vec {r}}= \dot{r}\vec{u_r}+r\dot{\theta}\vec{u_\theta}+r\sin\...
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Meaning of the differential entropy
The definition of differential (or continuous) entropy is problematic. As a matter of fact, differential entropy can be negative, can diverge and is not invariant with respect to linear ...
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Velocity to Acceleration negative line [closed]
Is the velocity line in below 0 is a different acceleration line?
For example from 0 - 6s and from 10 - 17s.
It has the same slope.
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How to calculate the rotation at a singularity?
An electrodynamics lecture asks me to prove that
$$
\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}+ \...
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Entropy as an exact differential
I have been trying to prove that $dS$ is an exact differential given this definition:
If you have a function of two variables $z = f(x, y)$ for which $dz$ can be expressed as:
$$dz = \left(\frac{\...
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Maxwell's relations and adiabats
I was trying to understand problem regarding finding the adiabatic modulus given the isothermal young's modulus. I'm still an amateur in thermodynamics.
I just didn't understand the final step where ...
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Weird derivative with respect to inverse temperature identity in Tong's statistical physics lecture notes
While reading David Tong's Statistical Physics lecture notes (https://www.damtp.cam.ac.uk/user/tong/statphys.html) I came across this weird identity in page 26 (at the end of the 1.3.4 free energy ...
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Tensor Index Manipulation
I am trying to study General Relativity and I thought about starting with some index gymnastics. I found a worksheet online and I am stuck with a simple problem. I have to prove that
$$\partial_{\mu} ...
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Equality of variables for small values of time, when the time derivative of the variables are equal to one another
It is given that $\frac{ds}{dt} = \frac{d\theta}{dt}$, i.e. the time derivative of s and $\theta$ are equal to each other.
Does it follow that for small values of $t$, $\Delta s ≈ \Delta \theta$? My ...
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How do extreme points work in Statistical Mechanics?
Suppose that I have an $S,V,N$ ensemble. Every variable is a function of the other variable: $U(S,V,N)$, $S(U,V,N)$, $V(S,U,N)$ and $N(S,U,V)$. The functions are everywhere differentiable. But there ...
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Why does $\vec{a}=\vec{\omega}\times \vec{r}$ as well as the velocity does?
Today I came in class and in one of the problems the teacher used $\vec{a}=\vec{\omega}\times \vec{r}$ which made me very confused because I don't know where it comes from, it seems pulled out of thin ...
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How does the covariant derivative satisfy the Leibniz rule?
In Carroll's "Spacetime and Geometry", he states on page 95 (section 3.2) that the covariant derivative, $\nabla$, is a map from $\left(k, l\right)$ tensor fields to $\left(k, l+1\right)$ ...
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How to evaluate the action of a fractional differential momentum operator?
I need to understand how a fractional operator works if before being applied on a test function it acts on another (well known)function:
$$(-i\hbar\frac{\partial}{\partial x}\cdot\delta(x))^\epsilon\...
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From where does the uncertainty formula come from? [duplicate]
The uncertainty formula is the one used in the laboratories to find the uncertainty of a variable. Say $X$ is a function of $Y$ and $Z$ such that $X=X(Y,Z)$ then it's uncertainty can be found with: $$\...
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Is the Lie derivative in a coordinate direction covariant?
Considering a partial derivative of a vector field $w^a$ in x-direction (also called here 1-direction) I can write it as $$\frac{ \partial w^a}{\partial x^1 } = \partial_1 w^a - \Gamma^a_{1c} w^c + \...
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How to understand the derivatives in wave equation?
I am looking at the derivation of the wave equation, but I am stuck on the math. Specifically, in the following:
How do they get the equivalence between $\frac{\partial}{\partial z} (\frac{dg}{du}) = ...
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Lagrangian - How can we differentiate with respect to time if $v$ not a function of time?
In the Lagrangian itself, we know that $v$ and $q$ don't depend on $t$ (i.e - they are not functions of $t$ - i.e., $L(q,v,t)$ is a state function.)
Imagine $L = \frac{1}{2}mv^2 - mgq$
Euler-Lagrange ...
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How does the divergence change under a change of frame (with geometric algebra)?
I'm trying to prove equations (85) and (86) from Hestenes' paper Gauge Theory Gravity with Geometric Calculus (ResearchGate version).
$$
\dot{\nabla}^\prime \cdot \dot{\underline{f}}(A) = J_f^{-1}[ (\...
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Variation of Torsion-Free Spin Connection
In the book 'Supergravity' by Freedman and van Proeyen, in exercise (7.27) it is written
To calculate [the variation $\delta\omega_{\mu ab}$ of the torsion-free spin connection], consider the ...
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