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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Covariant Derivative: What does changing direction mean in curved space?

I am on my way to general relativity, but I am struggling with the covariant derivative. At this point I am trying to ignore the spacetime character of the world i.e. I am trying to understand what ...
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In the equation: $a = dv/dt$ , is $dt$ the time taken to achieve that instantaneous acceleration?

If you solve for $dt$ from $a = \frac{dv}{dt}$ , is it the time taken to to achieved that instantaneous acceleration? $a$ : acceleration $v$ : velocity $t$ : time
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What does the first derivative of (2-norm) distance with respect to time tell us?

My basic physics' knowledge is a little rusty. My apologies in advance. I know that the first derivative of position or displacement with respect to time is the instantaneous velocity. Suppose I have ...
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When can I treat derivative as a fraction? (Brachistochrone)

My teacher was solving the Brachistochrone problem in class. She parametrized the required path with $x(y)$, then said $T=\int_0^Tdt=\int_{y_1}^{y_2}\frac{dt}{dy}dy=\int_{y_1}^{y_2}\frac{dy}{dy/dt}$. ...
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Total Differential [closed]

Assume that $$\delta Q = dU+pdV $$ is a total differential. Show that this assumption leads to the false Statement $$\left(\frac{\Delta p}{\Delta T}\right)_V = 0.$$ Can anyone give me a hint how to ...
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Gradient of a function vs. Divergence of a four-vector [closed]

Solving the gradient of a function $\phi$ i.e. $\partial_\mu$ is actually the gradient of the function where $\partial_\mu$ has one component of time and 3 component of vector in 4-vector and same is ...
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When the rate of acceleration changes it's sign how does the velocity change?

When the rate of acceleration changes its sign how does the velocity change? When another derivative of distance with respect to time is increased how does it affect factors like displacement and ...
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Why does $q(t) \to-i\hbar \frac{\delta}{\delta J(t) }$ for the generating functional of a perturbed harmonic oscillator?

When computing a generating functional, $Z[J]$, in terms of the generating functional of Green functions, $Z[0]$, in my lecturer's notes we reach the following terms: $$Z[J]= \mathcal{N} \int Dq \...
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Why $\Gamma^\mu_{\beta\delta;\gamma} =\Gamma^\mu_{\beta\delta,\gamma}$?

Gravitation Page 276 Exercise 11.3 solution indicated that $$\nabla_\gamma \nabla _\delta e_\beta =e_{\mu}\Gamma^\mu_{\beta\delta,\gamma} +(e_\nu\Gamma^\nu_{\mu\gamma}) \Gamma^\mu_{\beta\delta}$$ ...
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Why does the path modification ($\delta$) commute with the time derivative in the derivation of the Euler-Lagrange equation?

When deriving the Euler-Lagrange equation my notes bring the $\delta$ into the action integral, which is fine, which gives $$\delta S=\int_{t_0}^{t_1}dt \frac{\partial \mathcal L}{\partial q_i}\delta ...
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Partial derivative of a 4-velocity

Trying to do some basic manipulations with 4-vectors and I have a question about the proper (no pun intended) approach. It's probably easiest if we look at a simple example. So let's define a 4-...
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How to see that kinetic energy depends on generalised coor., gen. vel. and time?

Let $u_{ik}$ denote the $i^{th}$ component of the position vector of the $k^{th}$ particle. Then kinetic energy of the system of $N$ particles is given by: $$T=\frac{1}{2}\sum^N_{k=1}\sum^3_{i=1}m_k\...
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About Feynman rules and symmetry factor

I have two simple questions about Feynman rules for Lagrangian that have a derivative of fields. For example for Lagrangian in this link part a with derivative couplings. Is the vertex depend on ...
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Is Ricci's theorem can be simply deduced using covariant derivatives of fundamental tensors? [duplicate]

Well Ricci's theorem is given by: $$\mathrm{D}g_{ij}=\mathrm{D}g^{ij}=0$$ I was wondering that if the theorem can be proved using covariant derivatives of $\delta_i^k$, $g_{ij}$ and $g^{ik}$. I ...
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Total Derivatives and Thermodynamics

Question 1 What are the following quantities functions of: volume, pressure, temperature and mass? For I am very confused when I should be using $d$ or $\partial$ for my derivatives in thermodynamics....
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What did I do wrong? I got $\nabla\cdot \vec A \neq div \vec A $ [closed]

We know, that in orthogonal Curvilinear coordinate system: $$ \nabla =\sum_{i=1} ^{3}{\hat{e_i} \over h_i}{\partial \over\partial u_i} $$ Let $$\vec A=\sum_{i=1} ^{3} A_i \hat e_i$$ Now $$ \nabla ...
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Can I find the acceleration or velocity when my displacement-time graph is discontinuous?

Today, I encountered the problem where I was asked to find the velocity and acceleration from displacement-time graph but the displacement-time graph was discontinuous. So I am unable to find the ...
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Product rule for 4-vectors and derivation of 4-force form

In deriving the form for the 4-force in special relativity, we begin with $$\frac{d}{d\tau}p^{\alpha}=m\frac{d}{d\tau}u^{\alpha}=f^\alpha$$ where $\tau$ is the proper time, m is rest mass. Since $...
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Question about proportionally rules

I don't think context is needed but to make sure: I'm doing a homework exercise on binary system. P is the orbital period and E the energy of the system. The following is in the solution when trying ...
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What are all the gauge symmetries & derivatives of the QED lagrangian?

I find that the gauge symmetries of the lagrangian are a topic that gets obfuscated quite a bit. I'm trying to understand the big picture of this in QED. My understanding is that: Gauge derives its ...
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What is proper time, proper velocity and proper acceleration?

I am trying to derive the relativistic rocket equations found here [(4),(5),(6),(7),(8)] but I do not understand proper time, proper velocity and proper acceleration. Define a point $P$ with ...
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Lie derivative of the non-coordinate metric being 0

I'm trying to answer a question about a the Lie derivative of a metric in a non-coordinate basis. Here, the $C^{a}_{bc}$ are from the Lie derivative of one basis vector with respect to another, or ...
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Hermitian conjugate of momentum operator

We know that the momentum operator must be Hermitian since its eigenvalue gives the momentum which is measurable and hence must be real. Now, when the momentum operator is written in the form $$\...
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Which derivative to use in the change of metric tensor due to a gauge transformation?

I'm used to calculating the change in the metric due to a gauge transformation in the following way: The gauge transformation up to linear order is \begin{equation} x^\mu \rightarrow x' ^\mu =x^\mu ...
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Replacing infinitesimals with full vectors in a differential relation. Is it legit?

I'm reading Leonard Susskind's "Special Relativity and Classical Field Theory". On pg. 138 he generalizes a differential relation by replacing infinitesimals with full vectors like so: Is this ...
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1answer
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Basic index question on tensor calculus

after some calculations I obtain the next expression $$\lambda^{c}\nabla_{a}g_{bc}=\lambda^{c}{}(\lambda_{c}\nabla_{a}\lambda_{b}-g_{be} \Gamma^{e}_{ad}\lambda^{d}\lambda_{c}) $$ So my question is ...
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Relation bewteen to different operators in different charts

Let $\phi$ be a coordinate system and $\partial/\partial^{\mu}$ and $dx^{\mu}$ be the associated coordiantes bases. Then in the region covered by these coordinates we can define a derivative ...
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Can value of the variable be substituted in partial derivatives before taking the derivative?

I was going through the D'Alembert's solution for the wave equation using this pdf from University of British Columbia (UBC, Canada). Here's the link: https://www.math.ubc.ca/~ward/teaching/m316/...
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Identity of covariant derivative

I was reading about Einstein-Hilbert action, and in some point in this page they use this identity $$\sqrt{-g}A^{a}_{;a}=(\sqrt{-g}A^{a})_{,a}$$ I Know that $\nabla_{\sigma}g_{\mu\nu}=0$. And $g$ ...
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What is the covariant derivative of the Ricci Tensor?

How do I take the covariant derivative of the Ricci Tensor? Could someone be so kind as to give me the process of how it is done?
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Derivative of Lagrangian with respect to velocity

My question revolves around this lecture notes on page $109$ equation $(5.1.10)$. Let's stick to $\mathbb{R}^3$ and consider a particle in $3$-space with position vector $\mathbf{x} = (x, y, z)$. ...
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Calculate the expression of divergence in spherical coordinates $r, \theta, \varphi$

Hi this is my first question in [Physics.SE] I saw a lot of posts and I liked them. I hope that my question will be answered too. While I'm solving a problem in vector calculus. I recognized that I ...
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Confusion regarding the time derivative term in Lagrange's equation

I am solving a pendulum attached to a cart problem. Without going into unnecessary details, the generalised coordinates are chosen to be $x$ and $\theta$. The kinetic energy of the system contains a ...
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Basic question of General relativity about covariant derivative

I was reading the book of Wald on General relativity. And in the page number (33) he derives the equation for the action of $\nabla_{a}$ over a tensor of rank $(k,l)$. This is the equation (3.1....
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Fréchet derivative of a operator $E: H_{per}^{1}\left([0,L]\right) \longrightarrow \mathbb{R}$ [migrated]

Define the operator $E: H_{per}^{1}\left([0,L]\right) \longrightarrow \mathbb{R}$, given by $$E(u)=\frac{1}{2}\int_{0}^{L}(u_t^2+u_x^2+\frac{1}{2}(1-u^2)^2)dx,\; \forall \; u \in H_{per}^{1}\left([0,...
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Can we calculate centripetal acceleration by using this method $\frac{\mathbf v_2-\mathbf v_1}{T}$?

If we know the angle between two velocity vectors $\mathbf v_1$ and $\mathbf v_2$, and if we know the time $T$ it takes for the velocity to change from $\mathbf v_1$ to $\mathbf v_2$,then is it ...
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Divergence of a vector which has explicit and implicit position dependence

I am doing EMT and I am trying to calculate the divergence of this current density given as, $$\vec{J}(\vec{r}', t_r) = \vec{J}(\vec{r}', t - \frac{|\vec{r}-\vec{r}'|}{c})$$ for $\vec{r} = (x,y,z)$ ...
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Covariant derivative of a metric determinant

The covariant derivative of a metric is zero $g_{\alpha\beta;\sigma}=0$. Is the covariant derivative of a metric determinant zero following the assumption($g_{\alpha\beta;\sigma}=0$): $$ g_{;\sigma}=...
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What does $\frac{Dv}{Dt}$ and $\frac{D\rho}{Dt}$ notion mean here?

Momentum conservation: $$\rho\frac{Dv}{Dt}=\nabla\cdot\sigma+\rho g$$ Mass conservation: $$\frac{D\rho}{Dt}+\rho\nabla\cdot v=0$$ What does $\frac{Dv}{Dt}$ and $\frac{D\rho}{Dt}$ notion mean here ...
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1answer
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Confusion regarding the terms in the covariant derivative of a Tensor

I am learning General Relativity from Leonard Susskind's Lectures. In Lecture three, he introduces to covariant derivatives, and I understood it's meaning. But when he applies it to a Tensor, I am ...
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D'Alembert's principle derivation in Goldstein's Classical Mechanics book [duplicate]

(I could not find any answer to the following question in other related questions posted on SE, so asking it here.) In the derivation of D'Alembert's principle in his "book", Goldstein uses the ...
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Derivative of four-vectors [closed]

I have to take the derivative of $(P+2k-2q)^2$ with respect to $k^{\alpha}$. What I am doing is writing $(P+2k-2q)^2=(P+2k-2q)^{\alpha}(P+2k-2q)_{\alpha}$ Then its derivative comes to be $2(P+2k-...
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Inverse of the coordinate transform Jacobian

Assume a point's position is given by the coordinates $x_i$. Introducing a new set of coordinates $\Theta_i$, one can relate the differentials $d\mathbf{x}=(dx_1, dx_2, dx_3)$ and $d\mathbf{\Theta}=(d\...
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1answer
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Gauge covariant derivative and Leibniz rule

Let's say I've got 2 different fields $a, b$ and I want to compute its covariant derivative $D_\mu = \partial_\mu + iA_\mu^a T^a$ where $\{A_\mu^a\}$ is the set of gauge fields and $\{T^a\}$ the ...
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Why don't we see the covariant derivative in classical mechanics?

I am wondering why I have seen the covariant derivative for the first time in general relativity. Starting from the point that the covariant derivative generalise the concept of derivative in curved ...
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How to take the curl of the angular momentum operator?

I'm trying to show $$\nabla \times \vec{L} = \frac{1}{i}(\vec{r}\nabla^{2}-\nabla(1+r\frac{\partial}{\partial{r}})) $$ where $ \vec{L}\psi = \frac{1}{i}(\vec{r}\times \nabla)\psi$. I'm able to expand ...
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Motivating the definition of the gauge covariant derivative from E&M

Source: Faddeev and Salvnov's Gauge Fields page 4 "The EM field interacts with charged fields, which are described by complex functions $\psi(x)$. In the equations of motion the field $A_\mu(x)$ ...
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Can you use $a=$$\frac{\Delta v}{\Delta t}$ instead of $\frac{dv}{dt}$ to find instantaneous acceleration?

Can you use $\frac{\Delta v}{\Delta t}$ instead of $\frac{dv}{dt}$ to find instantaneous acceleration?
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What mean this momentum-derivative?

I'm working with quantum gravity. I have to make a Taylor-series. I got some help for this, but I have problem with understanding the formalism. So, I have the operator $A((P-p)^2)$, which needs to ...
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Covariant derivative of four-position

May someone confirm or deny that covariant derivative of four-position is just metric tensor? I mean: $\nabla_{\gamma}X_{\alpha} = g_{\gamma \alpha}$ When I try to rewrite it with base vectors it ...

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