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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Infinitesimally small time intervals

When saying that in a small time interval $dt$, the velocity has changed by $d\vec v$, and so the acceleration $\vec a$ is $d\vec v/dt$, are we not assuming that $\vec a$ is constant in that small ...
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Implications of Galilei-Invariance on a time-independent potential

I'm trying to compute a result shown in my classical mechanics lecture on my own. Namely, consider that a system composed of $n$ particles follows a law of force $m_k\ddot{\vec{x_k}} = \vec{F_k}(\vec{...
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Einstein field equations from covariant derivative of a general linear gauge transformation

A general linear transformation is given by \begin{align} \psi'(x) \to g \psi(x) g^{-1}, \end{align} The gauge-covariant derivative associated with this transformation is \begin{align} D_\mu \psi=\...
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Differentiation [closed]

Why is $$\frac{d}{dt}v^2=2v\frac{dv}{dt},$$ When: $$\frac{d}{dx}x^2=2x,$$ where $v$ is velocity? I don't understand why the variable $x^2$ has the derivative of $2x$, whereas the variable velocity has ...
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How to move a $d/dt$ inside a triple integral? [migrated]

I faced some equations while reading current density. $$\iint_S \vec J.d \vec S = - \frac{d}{dt} \iiint_V \rho.dV.$$ And then $$\iint_S \vec J.d\vec S = -\iiint_V \frac{\partial \rho}{\partial t}dV. $$...
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Explicit calculate covariant derivative for spinor field

I want to explicitly calculate the covariant derivative for spinor-fields for a given metric (to investigate the 1d dirac equation in curved spacetime): $\begin{align*} g_{\mu\nu}=\frac{L^2}{cos^2(\...
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7 answers
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Is every $dm$ piece unequal when using integration of a non-uniformly dense object?

When we want to find the total charge of an object or total mass, usually we start off with a setup such as: $$ m = \int dm \:\;\:\text{or} \:\;\:q = \int dq$$ in which we then use (and to keep it ...
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Why is instantaneous velocity tangent to trajectory?

Trajectory is the path of an object through space as a function of time. However, in many trajectory plots, when the movement is planar, a horizontal position axis and a vertical position axis are ...
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In Radius of Curvature calculation why do I have to assume $\text{d}^2x/\text{d}t^2=0$?

Recently, I was calculating the radius of curvature of projectile trajectories at certain points. There are two ways to do the same: Given the velocity and acceleration of the particle at some point, ...
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Fourier transformed quantity always representable by derivative?

Suppose we perform the following steps to go from an integral over the spatial dimension to one over the momentum: $$\int dx \,f(x) = \int d x\left(\int \frac{d k}{2\pi}\,\tilde{f}(k)\, e^{ikx} \, \...
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Commutator between covariant derivative and a field

I have field as an element of a Lie algebra as $\Phi = \phi^at^a$ and I want to calculate the commutator $$[D_{\mu}, \Phi],$$ with $$D_{\mu} = \partial_{\mu} + igA^a_{\mu}t^a,$$ the covariant ...
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Atomic force microscope tip stability

I want to ask why AFM (atomic force microscope) tips snap to contact (also called jump to contact). I know that a system is stable when the force from the cantilever (given by Hooke's law) is equal to ...
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1 answer
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Determine the meaning of a gradient of a graph [closed]

How do you determine the gradient of a graph in physics, such as how with a velocity-time graph, the gradient is acceleration. I want to know the general method for figuring out what the differential ...
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2 votes
1 answer
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Proof that the Euler-Lagrange equations hold in any set of coordinates if they hold in one

This is a question about a specific proof presented in the book Introduction to Classical Mechanics by David Morin. I have highlighted the relevant portion in the picture below. In the remark, he ...
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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, ...
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Higher dimension derivatives

In the case of higher dimensions (e.g. 4+1 dimensions) how would the 5 derivative ($\partial_5$) change? For example if $\\X=(x^{\mu},z)$, would the 5 derivative change as $$\partial_5\partial^5X=\...
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What is the meaning of $\nabla _{\mu}\nabla _{\nu}\phi(r)$ in general relativity?

I know the covariant derivative of a tensor is $$\nabla_{\mu} V_{\nu}=\partial_\mu V_\nu-\Gamma_{\mu\nu}^{\lambda}V_{\lambda}$$ Now I want to obtain $\nabla_{\mu}\nabla_{\nu}\Phi(x)$ where $\Phi(x)$...
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Taking the second time derivative of a scalar field

Given some scalar field $\phi(x,y,x,t)$, taking its first total derivative we get: $$\frac{d\phi}{dt}=\frac{\partial\phi}{\partial t}+\frac{\partial\phi}{\partial x}\frac{dx}{dt}+\frac{\partial\phi}{\...
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Question about Wald's example of a "derivative operator"

I am trying to study Wald's book on General Relativity. I thought the first two chapters were ok, but I'm completely stuck in chapter 3, on page 32 where he gives an example of a "derivative ...
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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....
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Finding minimum value of a function [migrated]

Consider $F$ is a function of $x$. To find the minimum value of $x$ which produces a minimum value of $F$, we are required to do $dF/dx = 0$. (Differentiate $F$ with respect to $x$) Why do we do this ...
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Standing Wave Equation: Why does assuming a small slope $du/dx$ not make $d^2u/dx^2$ negligible as well?

Referencing the above image, just change the label for $y$-axis to $u$-axis.^ Following the derivation of the standing wave equation: https://www.youtube.com/watch?v=IAut5Y-Ns7g&t=1324s So if ...
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1 vote
1 answer
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Simple difference between module of velocity and time derivative of module of position [duplicate]

What is the conceptually difference between the two: $$\frac{d|\vec{r}|}{dt}=\frac{\vec{r}\cdot\frac{d\vec{r}}{dt}}{|\vec{r}|}\neq|\dot{\vec{r}}|\equiv \bigg|\frac{d\vec{r}}{dt}\bigg|$$ ...
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2 answers
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Why are formulas in Physics represented in form of differentiation?

Mostly formulas are represented in differential form as we learn more about the concepts of physics , for ex. $I = \frac {q}{t}$ is also written as $\frac {dq}{dt}$ A explanation would help .
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Variation of the Lagrangian

In Tong's QFT notes at the bottom of page 14, it is claimed that if a change $x\mapsto x-\epsilon$ is made, the Lagrangian changes in the following way: $$\mathcal L(x)\rightarrow \mathcal L(x)+\...
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4 votes
1 answer
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What does $\delta/\delta t$-derivative represent in tensor calculus?

Some texts, such as Pavel Grinfeld's, talk about a $\delta/\delta t$-derivative whose role (in trajectory analysis of particles using tensor calculus) is pretty obscure to me. For example, the ...
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10 votes
7 answers
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What is the instant velocity? [duplicate]

The velocity is the variation rate of the position correct? So does it make sense to talk about velocity without time?
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Currently self-studying QFT and The Standard Model by Schwartz and I'm stuck at equation 1.5 in Part 1 regarding black-body radiation

So basically the equation is basically a derivation of Planck's radiation law and I can't somehow find any resources as to how he derived it by adding a derivative inside. Planck says that each mode ...
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1 answer
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Non-parallel light diffraction

Does light (and in general any kind of wave) diffract only when the wave fronts are parallel? Like if you did the double slit experiment when the waves were coming from a point source close to the ...
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3 votes
1 answer
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What are the spaces in which quantum fields belong and how does that affect the hermitian conjugate of $\partial_{\mu}$?

In this post, I claimed in my answer that $\partial_{\mu}^{\dagger}=\partial_{\mu}$. The reason I claimed that is because $D^{\dagger}_{\mu}=\partial_{\mu}-iqA_{\mu}$ and I assumed that $$(\partial_{\...
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4 votes
1 answer
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Generalizing Fermi-Walker Derivative/Transport to General Vector Bundles

The usual definition of the Fermi derivative (eg as given in Hawking and Ellis) is to consider a Lorentzian manifold $(M,g)$ and a unit timelike curve $\gamma$, a smooth vector field $X$ along $\gamma$...
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1 vote
1 answer
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Derive interaction lagrangian for KG equation in QED

The free-field KG lagrangian density for complex scalar field is given as $$\hat{\mathcal{L}}_{\text{KG}}=(\partial_\mu\hat{\phi}^\dagger)(\partial^\mu\hat\phi)-m^2\hat{\phi}^\dagger\hat{\phi}$$ By ...
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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 ...
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2 answers
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Can velocity be defined as the rate of change of displacement and the rate of change of position?

Can we have velocity = $\dfrac{ds}{dt} = \dfrac{dr}{dt} = v(t)$? Can we define as they should be the same in any frame of reference, say we have any $1$ Dimensional frame $F_1$, we will have a ...
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8 votes
2 answers
712 views

Physical significance of metric compatibility

When we try to construct a covariant derivative, we impose several conditions on it so that the resulting derivative is unique. However, I can't make sense of the condition of metric compatibility. I ...
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1 vote
1 answer
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Srednicki 11.3 part e) Finding the maximum energy for the electron

In part e you are asked to find the differential decay rate, \begin{equation} \frac{d\Gamma_{\mu^- \rightarrow e^- \bar{\nu_e} \nu_\mu}}{dE_e} = \frac{mG_F^2}{ 4\pi^3 } \big( mE_e^2- \frac{4}{3}...
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2 votes
3 answers
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Derivation of entropy, I don't understand the relation $ \frac{\partial S_2}{\partial E_1} = -\frac{\partial S_2}{\partial E_2} $

My course guide gives the following derivation for change in entropy w.r.t. energy, where I don't understand a step: \begin{align} E & = E_1 + E_2 \\ S & = S_1 + S_2 \\ S(E,E_1 ) & = S_1 (...
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Velocities - Equation 1.46 of Goldstein 3rd edition

In his derivation of the Euler-Lagrange equations from D'Alembert's principle, Goldstein uses the parametrization (equation 1.45') $$\displaystyle{\vec{r_i}=\vec{r_i}(q_1,q_2, ..., q_n, t)}\tag{1.45'}$...
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4 votes
3 answers
197 views

Ricci Identity with Torsion Proof

In the notes I'm using for General Relativity, the author begins their proof of the Ricci identity with torsion by writing $$\nabla_{[\mu}\nabla_{\nu]}Z^{\sigma}=\partial_{[\mu}(\nabla_{\nu]}Z^{\sigma}...
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0 answers
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Can we define $\text dW$? [duplicate]

I am currently taking applied thermodynamics at my university, and for the definition of entropy this is the formula used in the book (Thermodynamic for Engineers by Moran, Shapiro, Boettner, Bailey): ...
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1 answer
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Rigorous treatment for continuous mass systems

I would like to ask if anyone knows an accessible, yet rigorous way of passing from a discrete system of mass-points to a continuous mass system. For instance, we clearly know how to define the ...
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1 vote
1 answer
26 views

Convective derivative N-S

This is probably an easy answer, but I've not been able to find it yet - Why in some formulations of the N-S equations (for example here https://www.grc.nasa.gov/www/k-12/airplane/nseqs.html), is the $...
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2 votes
1 answer
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Notation and Terminology Questions from Schwartz' QFT Book

I am finding some of the notation confusing in Chapter 3 on Classical Field Theory in Schwartz' QFT book a bit confusing. First off, on page 34 he defines a translation of a field to first order as $$...
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0 answers
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Mayard's mechanical differentiator

I've recently been reading an article on a mechanical diferentiator (Nouvelles solutions de calcul grapho-mécanique. Dérivographe et planimètre). The author describes a mechanical device, which, given ...
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-1 votes
1 answer
70 views

Why can you remove constant when taking the derivative? [closed]

If the derivative of a constant is 0, why can we just remove this constant when differentiating? eg. If d/dx(3x^2+5x+1), can we write this as d/dx(3x^2)+d/dx(5x)+d/dx(1). If so, what allows us to do ...
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0 votes
1 answer
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Step in derivation of Lagrangian mechanics

There is a step in expressing the momentum in terms of general coordinates that confuses me (Link) \begin{equation} \left(\sum_{i}^{n} m_{i} \ddot{\mathbf{r}}_{i} \cdot \frac{\partial \mathbf{r}_{i}}{\...
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1 vote
2 answers
138 views

Why can I write $\frac{d}{dt}=\frac{d}{dt'}\frac{dt'}{dt}+\frac{d}{dx'}\frac{dx'}{dt}$?

I’m dealing with a Lorentz invariance problem, and in one of the solutions I’ve seen to prove the wave equation the term above was used. However I don’t really understand why it can be written that ...
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1 vote
2 answers
81 views

Maxwell's eq-meaning of del's cross and dot product?

In maxwell's eq there is del whose cross and dot products exist. So what is del in cross vs dot product. What's the difference when it's just a partial differential operator.
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2 votes
1 answer
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Does the expression "$𝑑𝑠^2$..." mean the same thing as "$\Delta 𝑠^2$... "?

I reviewed this question but sometimes I'm unsure about delta versus differential notation. Does the expression "$ds^2=-c^2dt^2+a^2(t)[dr^2 + S_k^2(r)d\Omega^2 ]$" mean the same thing as &...
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1 vote
1 answer
35 views

Energy change under point transformation

How do the energy and generalized momenta change under the following coordinate transformation $$q= f(Q,t).$$ The new momenta: $$P = \partial L / \partial \dot Q = \partial L / \partial \dot q\times ...
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