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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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233 views

What is the time derivative $\frac{d}{dt}(\exp(\hat A))$ of operator exponential $\exp(\hat{A})$? [duplicate]

Assuming that the operator $\hat{A}$ does not necessarily commute with $d \hat A/dt$, what is $$\frac{d}{dt}(\exp(\hat A))~?\tag{1}$$ Is it $\exp(\hat A(t)) \frac{d \hat A}{dt}$ or $\frac {d\hat A}{dt}...
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1answer
92 views

When can we raise lower indices on “nontensors” as described in Dirac's book *General Theory of Relativity*?

This is a follow on to my previous question: Dirac's book *General Theory of Relativity*: Doesn't this show the partial derivative of the metric tensor is zero? I should not have made that ...
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1answer
67 views

What is the industry standard definition of $\nabla\cdot\mathfrak{T}$ (del dot a tensor)? Re: MTW

In chapter 3 of MTW's Gravitation using the example of a rank-3 tensor $\mathfrak{S}$ they define $$\text{(divergence of }\mathfrak{S}\text{ on the first slot)}\equiv{\nabla\cdot\mathfrak{S}}$$ ...
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Why an elementary work is written $\delta W$ instead of $dW$? [duplicate]

Why an elementary work is written $\delta W$ instead of $dW$? For example, it's often written $$\delta W=F\cdot dr$$ if $dr$ is the elementary displacement. Why don't we write as usual $dW=F\cdot dr$?
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130 views

Dirac's book *General Theory of Relativity*: Doesn't this show the partial derivative of the metric tensor is zero?

See the bold text for my question. This question regards Dirac's General Theory of Relativity page 13. In his demonstration that the length of a vector is unchanged by parallel displacement Dirac ...
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1answer
127 views

Why $(\frac{\partial S}{\partial T})_P=(\frac{\partial S}{\partial T})_V+(\frac{\partial S}{\partial V})_T(\frac{\partial V}{\partial T})_P$?

In the thermodynamics book (Adkins) I'm using, the following relation is cited without reference but I am not sure where it comes from. $$\left(\frac{\partial S}{\partial T}\right)_P=\left(\frac{\...
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1answer
103 views

Tensorality of the Lie Derivative and $i([X,Y]) = [L(X),i(Y)]$

I'm trying to understand the equation following (2.15) on p.9 of Blau's Symplectic Geometry and Geometric Quantization. For two vector fields $X,Y$ on a symplectic manifold $M$ we are told one ...
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1answer
86 views

Ambiguous curl for a vector field

I'm trying to work out how to explain why the curl/divergence for the following image is ambiguous without giving the explicit vector function for the following: E has an ambiguous divergence and I ...
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0answers
160 views

Thermodynamics and differential forms

In Potter's Thermodynamics: Demystified (page 68), the author wrote: Since only two independent variables are necessary to establish the state of a simple system, the specific internal energy can ...
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1answer
198 views

How come $\frac{d}{dt}\left(\frac{\partial {r_i}}{\partial {q_j}}\right) = \frac{\partial {\dot r_i}}{\partial {q_j}}$ in Lagrangian mechanics? [duplicate]

It is written in the Goldstein's Classical Mechanics text that $$\frac{\mathrm d}{\mathrm dt}\left(\frac{\partial {r_i}}{\partial {q_j}}\right) = \frac{\partial {\dot r_i}}{\partial {q_j}}=\sum_k \...
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1answer
122 views

Why aren't Christoffel symbols tensors? - asked from a fibre bundle perspective

I've been reading about connections on fibre bundles recently and it's made me think about the exact nature of the Christoffel symbols in GR. If we have a vector bundle $E$ over $M$ and put a ...
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1answer
90 views

Why is $\partial_{\mu}x^{\nu} = \delta^{\nu}_{\mu}$?

In Blundell's book on QFT, one can find the following Is this because of: $$\partial_{\mu}x^{\nu} = \partial_{\mu}x^{\nu^{'}} \partial_{\nu^{'}}x^{\mu}$$ $$\partial_{\mu}x^{\nu} = \Lambda_{\mu}^{\...
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196 views

How do “divergence” and “curl” relate to the three states of matter?

A fluid is said to have divergence (the ability to flow) and curl {the ability to rotate). Do these two characteristics fully define a fluid, or are there other important properties that I am missing? ...
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141 views

Question about derivation of four-velocity vector

In order to describe a notion of rate of change of positon, in four-dimensional spacetime, we have to introduce the concept of four-velocity. So, consider the following: For a massive particle ...
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2answers
125 views

Physical Interpretation of d'Alembert Operator

$$\mathop{{}\Box}\nolimits=\frac{1}{c^2}\frac{\partial^2}{\partial t^2}-\mathop{{}\bigtriangleup}\nolimits$$ is the d'Alembert-operator. It seems to consist of an oscillation and a diffusion. Is there ...
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1answer
58 views

Double divergence of stress tensor for migration flux

I am looking to calculate migration as a function of time using equation in Image 1. SigmaP is the total particle stress tensor in the cylindrical coordinates (r, theta, z). I am only interested in ...
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1answer
390 views

Commute covariant derivatives of spinors

Consider a spinor field $\psi$ on a general smooth Lorentzian manifold. Let $\Sigma_{ab}$ be a matrix representation of the Lorentz group, and let Greek/Latin letters represent world/Lorentz indices. ...
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2answers
133 views

Confused with derivative and partial derivative

suppose $x=f(t)$ with a constant acceleration. Then does $\frac{\text d x}{\text d t} = \frac{\partial (x)}{\partial(t)}$ since the position in $x$ only changes with time? Then the acceleration in ...
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1answer
76 views

Why is velocity mathematically describes as a division? [duplicate]

I want to know why, in kinematics, is velocity described as $v = \frac{\Delta x}{\Delta t}$, and why it is not described as any other expression (like a multiplication), why does a division is the one ...
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2answers
102 views

Theoretical definition and pratical mesurement of differential cross section

In Sakurai's book, the definition of differential cross section is: $$d\sigma/d\Omega= transition \;rate / probability\; flux $$ However this def doesn't contain any information about the thickness ...
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1answer
93 views

Relation between curl and gradient [duplicate]

I need to prove the following relation (with vector $\mathbf{V}$) : $$(\mathbf{V} \cdot \nabla)\mathbf{V} = \frac{1}{2}\nabla (\mathbf{V}^2)+(\nabla \times \mathbf{V}) \times \mathbf{V}\quad\quad\...
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4answers
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With what velocity are we moving along the time dimension?

Does the question make sense? Velocity along time axis means $v_t=\mathrm dt/\mathrm dt$? If it doesn't, please explain where the flaw is. Taking time as measure like length? Or do we need to ...
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1answer
32 views

Vector Identity For Electrostatics

I am reading about electrostatics and came across this vector identity when discussing the $D$ field: $$\frac{\nabla k_{e}}{k_{e}} = \nabla \ln (k_{e}).$$ I have not seen this identity before and ...
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1answer
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Physical significance of divergence

In my textbook They considered a parallelopiped $ABCDEFGH$ with sides $dx,dy,dz$ parallel to $x,y,z$ axis respectively $\vec V$ represents the vector velocity of the fluid at the centre $P$ of f ...
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1answer
45 views

Where is the error in this calculation of net curl for simple magnetic field?

I wasn't sure whether to post this on MSE, but PSE seems more appropriate. Let B be a static magnetic field in spherical coordinates, defined as $B=r\hat{\theta}$. Then, it's curl is $$\nabla \times ...
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1answer
42 views

Operator $A$ only act on the neighboured state or operator but not the entire expression?

In state vector formalism $A|\psi(x)><u(x)|=(A|\psi(x)>)<u(x)|$, where $A$ only act on $|\psi(x)>$ However, in terms of wave formalism, suppose $A$ is the well known $\frac{d}{dx}$. ...
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179 views

D'Alembert Operator

In which book or where can I find the derivation of the d'Alembert operator? \begin{equation} \Box \psi= \frac{1}{\sqrt{-g}}\partial_\mu \left( \sqrt{-g}\partial^\mu \psi \right) \end{equation}
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197 views

Is $\nabla=\nabla'$? Nabla operator acting on $r^n$

I have been taught that $$\nabla r^n =\text{gradient of }r^n =n r^{n-1}\ \hat{\boldsymbol r}$$ but in introduction to electrodynamics by Griffith (4th edition) on page 173, $\nabla' r^n$ is given by $-...
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1answer
125 views

Taylor expansion of scalar fields [closed]

Starting of with electrodynamics I have to compute the taylor expansion around $\vec{r} = 0$ of $\psi (\vec{r}) = |\vec{r} - \vec{r_0}|^{\frac{3}{2}}$ where $\vec{r_0}$ is a constant vector up to ...
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Have fractional order differential models been explored as an alternative to standard gravitational field theory?

Since Einstein introduced his field equations and general theory of relativity, experimental evidence, at least on the cosmic scale has repeatedly supported the theory. Nevertheless, many seeking to ...
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1answer
71 views

Planck Blackbody Radiation: Is this an error in the textbook?

the textbook I am reading describes two forms of equations of Blackbody Radiation. $$d\rho(\nu, T) = d\rho_\nu(T)d\nu = \frac{8\pi h}{c^3}\ \frac{\nu^3d\nu}{e^{h\nu/k_BT}-1}\ . $$ Substituting $ c = \...
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1answer
60 views

Dot product in cylidrical coordinates

I'm given the vector: $$\vec{V}{(r,θ,z)}=\frac{1}{r}\hat{e_r} + (r\cosθ)\hat{e_θ}+\frac{z^2}{r^2}\hat{e_z}$$ I want the scalar product ${\vec{\nabla}}\cdot{\vec{V}}$ We know that in cylindrical ...
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1answer
52 views

Derivative of wavefunction of a quantum system

If $\psi(x)$ represents the wavefunction of a 1D quantum system, it satisfies the Schrodinger equation, has a unit norm, and $\lim_{x\rightarrow \infty }\psi(x)=0.$ Then is it true that $\lim_{x\...
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1answer
65 views

Gauge covariance of the magnetic momentum operator

In the book about Schrödinger Operators by Cycon et al. there is a step in their calculations I don’t understand. When I pick two vector potentials $A_1$ and $A_2$ such that their curl (i.e. the ...
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3answers
331 views

How to determine the direction of instantaneous acceleration in a 2D motion? [duplicate]

How do we determine the direction of instantaneous acceleration when the body is moving in a plane (or a 3D space)? This question has been truly bothering me for nearly two weeks. I looked it up, ...
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1answer
86 views

Abuse of Calculus [duplicate]

I'm following Professor R. Shankar's Fundamentals of Physics course on YouTube. There I saw him doing manipulations of Calculus I never saw before. Here it goes, $$\newcommand\deriv[2]{\frac{\mathrm ...
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110 views

Uncertainty calculation - when to use absolute value bars?

I'm asking this because I saw (at least) two questions here on this Stack that seemed very similar and caused the same confusion to me in reading the answers to both. Suppose we have a formula: $A = ...
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2answers
212 views

Is $∂_\mu + i e A_\mu$ a “covariant derivative” in the differential geometry sense?

I have heard the expression "$∂_\mu + i e A_\mu$" referred to as a "covariant derivative" in the context of quantum field theory. But in differential geometry, covariant derivatives have an ostensibly ...
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0answers
233 views

Schwarzian derivative from conformal factor

Suppose I have a 2D Lorentzian conformally flat metric $$ ds^2 = -\Omega(u, v) du dv.$$ I consider a conformal field theory whose stress-energy tensor $T_{ab}$ is known on the flat metric $$ds^2 = -...
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1answer
545 views

Generalized divergence of tensor in GR

Although I've forgotten the proof (and cannot find it in, say, Carroll's book), the following formula holds for the covariant divergence in general relativity: $$\nabla_{\mu} A^{\mu} = \frac{1}{\sqrt{...
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1answer
3k views

How to calculate the jerk from acceleration data?

I have speed data from the GPS transmitter of a Truck which reports the speed of the vehicle at a fixed time interval. I can calculate the acceleration/deceleration of the truck by doing a $\frac{v_2-...
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2answers
147 views

Density of particles in the Lagrangian description of fluid flow

I am trying to learn about the material (or particle) derivative in fluid dynamics. I was looking through this explanation, and they mention things at the beginning and the end that confuse me. ...
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1answer
234 views

Covariant Derivative in QCD: How does it act on gluons?

Let the covariant derivative be $$D_\mu = \partial_\mu + \text ig\ A_\mu^a t^a,\quad a=1,\ldots,8$$where $g$ is the bare QCD coupling, $A_\mu^a$ are the eight gluon fields and $t^a=\tfrac{1}{2}\lambda^...
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2answers
76 views

Acceleration in a non-inertial reference frome - derevation

The general velocity equation for a point B in on body rotating and translating about point A with respect to the inertial reference frame say 'xyzo' can be expressed as, $\vec{r_{B/o}} = \vec{r_{A/o}...
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1answer
201 views

Derivation of curl of magnetic field [closed]

I am having trouble in one part of derivation of curl of magnetic field, from Biot-Savart law. The attached picture is from Griffiths - Introduction to Electrodynamics. I got all the parts, but only ...
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1answer
266 views

Can Yang-Mills field strength be defined as covariant derivative squared?

In Yang-Mills theory the field strength tensor $F_{\mu \nu}$ can be calculated as $$ \begin{equation} F_{\mu\nu} \equiv \frac{i}{g} [D_\mu,D_\nu] = \partial_\mu A_\nu - \partial_\nu A_\mu -ig[A_\mu,A_\...
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1answer
58 views

Need help creating an equation for the time it takes for a rocket to deaccelerate (accounting for fuel mass lost)

I'm doing a Math Internal Assesment for school where I'm trying to create a model for a rocket's suicide burn. I've gotten stuck in one of the equations I need for it. I need to find the time it would ...
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1answer
165 views

Calculating surface gravity

I have some trouble with understanding how surface gravity/Killing horizon equation works, for example in following form: $$κ^2=-\frac 12 (\nabla^aK^b)(\nabla_aK_b)$$ with Killing vector $K$. I'm ...
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3answers
32 views

Why the acceleration is specified if I know the coordinates and velocity?

And I don’t understand why the acceleration can be specified if we know the coordinates and velocity
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36 views

Non-differential total exact element notation question [duplicate]

I have questions regarding the notation in thermodynamics books of "d bar" (instead of delta) for the non-differential total exact elements like for work $\delta W$. When did it appear? Where are ...