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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What is the instant velocity?

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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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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2 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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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
683 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 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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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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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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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
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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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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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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
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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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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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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
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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
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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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Four-vector differentiation (E-M Euler-Lagrange eq.)

$$\partial_{\mu} \frac{\partial(\partial_{\alpha}A_{\alpha})^2}{\partial(\partial_{\mu}A_{\nu})} = \partial_{\mu}\left[2(\partial_{\alpha}A_{\alpha})\frac{\partial(\partial_{\beta}A_{\gamma})}{\...
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2 votes
1 answer
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How is it that adding a random field to the partial derivative results in a tensorial operation?

We know that the partial derivative of a tensor is not a tensor. But how is this problem fixed by adding to the partial derivatives, a field of Christoffel symbols? Christoffel symbols are a ...
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1 vote
2 answers
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When setting up 2nd order first derivative approximations in a finite differencing scheme, why are these equations equivalent?

In approximating a first derivative term (assuming $\delta z$ is the distance between two spatial grid points) using a finite differencing scheme I came up with these basic equations: $$\phi \frac{\...
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What do you call $ \frac{d^2 r}{dt^2}$ in polar coordinates? [duplicate]

In polar coordinates, one finds centripetal acceleration as: $$ a_c = \frac{d^2 r}{dt^2}- \frac{v^2}{r}$$ Where $|r|$ is distance from center to particle, $v$ is tangential velocity. My question is ...
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2 votes
2 answers
204 views

Exponential of an operator shifted by the derivative operator

Let $p(x)$ and $f(x)$ be sufficiently smooth functions and $D=\frac{d}{dx}$. It is easy to show that $$e^{p(x)D}f(x)=f(e^{p(x)D}x).\tag{1}$$ If $p(x)=a \in \mathbb{R}$ , we have the shift operator as $...
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2 votes
1 answer
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How to find covariant derivative of a contravariant tensor?

Let I have a contravariant tensor $A^\alpha$, I want to find covariant derivative of the contravariant tensor, From the transformation of the contravariant tensor ($A^\alpha=\partial_\gamma x^\alpha A^...
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-3 votes
2 answers
107 views

Explain this equation mathematically

$$\Bigl( \frac{\partial S}{\partial T} \Bigr)_H = \Bigl( \frac{\partial S}{\partial T} \Bigr)_M + \Bigl( \frac{\partial S}{\partial M} \Bigr)_T \Bigl( \frac{\partial M}{\partial T} \Bigr)_H$$ How can ...
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0 votes
3 answers
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How does the $\partial _{\mu} (\frac{\partial L}{\partial [\partial _{\mu} \phi]})$ term expand into a sum?

From QFT Demystified page 31: This term is from the Euler-Lagrange equation of a scalar field. How does this expand into a sum? Do we just sum over all $\mu$ from $\mu =0$ to 3, or are we supposed to ...
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Velocity gradient in a liquid

When we consider the motion of fluid in terms of many thin layers sliding over each other , we say that layer at a top of a layer forces it to move forward while layer below a layer forces it to move ...
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2 votes
4 answers
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Why do we use different differential notation for heat and work?

Just recently started studying Thermodynamics, and I am confused by something we were told, I understand we use the inexact differential notation because work and heat are not state functions, but we ...
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1 vote
0 answers
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Are covariant derivative on associated bundles exterior covariant derivatives?

The gauge covariant derivative we encounter in gauge theory $D\psi = d\psi + A\wedge \psi$ is a covariant derivative on the associated vector bundle, right? Here $\psi$ is the matter field, $A$ the ...
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In Griffiths' Electrodynamics, how does he calculate the contribution of two opposing faces of infinitesimal rectangular volume to a surface integral?

In appendix A of Griffiths' Electrodynamics, he sketches proofs of certain theorems of vector calculus. My question is related to this question, indeed the latter concerns the exact passage of the ...
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2 votes
1 answer
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Meaning of the transpose of a gradient

Sometimes I encounter PDE's with a term like this $\nabla \cdot c(\nabla \textbf{v} + (\nabla \textbf{v})^T)$ An example are the Navier-Stokes equations. Oftentimes this can be further simplified to $...
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3 votes
2 answers
202 views

Is MTW's covariant commutator $\left[\nabla_{a},\nabla_{b}\right]$ really the same thing as their vector field commutator $\left[a,b\right]$?

I now realize my question can be stated very concisely. In Chapter 11 of MTW, will the meaning be changed if in every instance we make the replacement $$\left[\mathbf{a},\mathbf{b}\right]\mapsto\left[\...
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3 votes
2 answers
276 views

Derivatives of exponential operator

I'm reading the paper (eq.(14) and eq.(10)) and got curious how the paper uses this equation: $\frac{\partial}{\partial c}\exp(-i\Delta t (X+cY)) = \exp(-i\Delta t (X+cY))(-iY\Delta t + \frac{\Delta t^...
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1 vote
1 answer
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Covariant derivative with an upper index in terms of Christoffel symbols

I have encountered expression $$\frac{1}{2}\left(2 \dot{g}_{\mu}{}^{\lambda ; \mu}-\dot{g}_{\mu}{}^{\mu ; \lambda}\right)$$ in a GR paper. Here we assume to be working with the de Sitter metric $g$ ...
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Why while solving some specific problems like this we need the second-order variation like this one?

Consider a string which is fixed at both ends and we are giving it a small amplitude by tauting it and moving to and from. If someone wants to derive the wave equation of the string, they would first ...
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Divergence of 4-velocity tensor

Today my professor told us that in Robertson-Walker metric the divergence of the 4-velocity of the element of the cosmic fluid $u^a_{\,;a}$ can be calculated with: $$u^a_{\,;a}=\frac{1}{\sqrt{-g}}(\...
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4 votes
5 answers
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Problem understanding the use of differentiation

I am new to differentiation. Our physics teacher gave us this example problem: The radius of a sphere is continuously increasing at the rate of 1 m/sec. Find the rate of change of the volume of the ...
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4 votes
1 answer
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Covariant derivative of the vielbein determinant

The vielbein postulate says that $$\nabla_\mu e_v^{\,a}=\partial_{\mu}e_\nu^{\,a}+\omega_{\mu\,\, b}^{\,\,a}\,e^b_\nu-\Gamma^\sigma_{\mu\nu}\,e^{\,a}_\sigma=0.$$ $\nabla$ is the coordinate covariant ...
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1 vote
3 answers
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Differentiating between tensors of different ranks

In my course on tensors matrices have been given as an example of a 2nd rank tenor, as they involve two indices, and similarly a vector as a 1st rank tensor. As it is possible to have a vector space ...
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MTW Box 10.1 E Additivity for covariant differentiation: Is this really a comprehensive proof?

This question is a prelude to another question I intend to ask. The following is from MTW Box 10.1 E. Additivity for covariant differentiation: E. In the real physical world, be it Newtonian or ...
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2 votes
1 answer
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Confusion regarding 4-Velocity Derivative Identity (for conservation of energy momentum tensor) in Carroll's Spacetime and Geometry

During Carroll's discussion of the energy-momentum tensor for a perfect fluid (page 36), he writes out that its divergence should be zero. He then expands this as follows: $$\partial_\mu T^{\mu\nu} = \...
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Covariant derivative, directional derivative, and curvature tensor

I'm confused about how to connect those three things together, so hopefully the question doesn't end up vague. The main problem is understanding how the curvature tensor is a commutator of covariant ...
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Find the curl if the vector field depends on a parameter

Given the following vector, \begin{align} F(x(t),y(t),z(t)) &= \begin{bmatrix} \omega_1^2 x_o\cos(\omega_1 t) \\ \omega_2y_0\sin(\omega_2 t)\\ 0\\ \...
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3 answers
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Limit of $d\rightarrow 4$ of a function in Peskin & Schroeder

In Peskin & Schroeder section 12.1 equation 12.15 we compute the function $$ \frac{-3\lambda^2}{(4\pi)^{d/2} \Gamma(\frac{d}{2})}\frac{(1-b^{d-4})}{d-4}\Lambda^{d-4} $$ Now when we take the limit $...
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