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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32 views

Finding the maximum electric field strength above a ring with a hole in the middle

I'm doing a problem (not homework, by the way) which asks for the electric field strength on the axis of symmetry a distance $x$ above the centre of a circular disc, which has uniform surface charge ...
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65 views

Intuitive Definition of Curl and Stokes' Theorem

I am asking this question on SE Physics because it's about an intuitive derivation, not a rigorously made proof of a theorem. Reading Mathematical Methods for Physics and Engineering by Riley, et al. (...
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Is there any difference in superscript and subscript notation in finite difference method

Is there any difference in superscript and subscript notation in the finite difference method? I see the same paper use (superscript for $x$ and superscript for $y$ notation) and (subscript for x and ...
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Two total differentials with equal variable differentials. Why coefficients in front of differentials are equal?

Could you prove that inference like that is valid: $$(1) \left\{ \begin{array}{c} dU=T dS-pdV \\ dU=\frac{\partial U}{\partial S}dS+ \frac{\partial U}{\partial V} dV \end{array} \right. \implies \...
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Does covariant derivative include magnitude change of a vector as well as direction change of the same vector?

Does covariant derivative include magnitude change of a vector as well as direction change of the vector? In some explanations I followed I have not noticed mentioning of magnitude change along with ...
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Planck's law question

I was reading Wikipedia article about Planck's Law and I wanted to make the same graph as here. I took this equation $$ B(\nu,T)=\frac{2h\nu^3}{c^2}\frac{1}{e^{\frac{h\nu}{kT}}-1}$$ but I got ...
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When is the swapping of derivative $d$ and gradient $\nabla$ valid?

In my plasma physics course, when studying the effects of magnetic mirrors, if we consider a magnetic field that primarily points in the $z$ direction as shown below: Picture source: F. Chen - ...
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How is this basic equation of a Timoshenko beam derived?

I am trying to learn how to model a Timoshenko beam which is described here: https://en.wikipedia.org/wiki/Timoshenko%E2%80%93Ehrenfest_beam_theory There are a few things I can't understand but the ...
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How do you differentiate this differential equation? [closed]

I have to differentiate this equation (Gravitational force between N-Bodies) $\begin{align} \frac{d^2}{dt^2}\vec{r_i}(t)=G \sum_{k=1}^{n} \frac {m_k(\vec{r}_k(t)-\vec{r}_i(t))} {\lvert\...
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101 views

Derivative as a fraction in deriving the Lorentz transformation for velocity

Consider a frame $S$ and $S'$ which is coincides at $t=0$ and then $S'$ starts moving with velocity $v$ in $+x$ direction. By Lorentz transformation equation, \begin{align} x'&=\gamma(x-vt) \\ ...
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Box size $R$ as a thermodynamic variable vs position radius $r$ inside the box

Titles are difficult but I hope I can do a better job in the text. I am working with a spherical box where I want to have the box's radius $R$ and temperature at the wall $T$ to be the main ...
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373 views

Do partial derivatives of different coordinate systems commute?

Consider an arbitrary set of coordinates $x^\mu$ and another set of coordinates $y^{\mu}$, which is a (lorentzian) transformation from $x^\mu$ given by $y^\mu = f(x^\mu)$. So I want to know whether $\...
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Hodge Laplacian and scalar

I'm reading Nakahara GEOMETRY, TOPOLOGY AND PHYSICS now.Then, Hodge Laplacian is given by \begin{align} \Delta=(d+d^{\dagger})^2=dd^{\dagger}+d^{\dagger}d \end{align} For example, we consider 0-form $...
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Is $\frac{dE}{dt}=0$ in an accelerating particle’s instantaneous rest frame?

My special relativity book uses an argument that involves $\frac{dE}{dt}=0$ in an accelerating particles rest frame (to show a force parallel to a particles velocity is parallel in all frames). ...
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If the displacement of an object is not differentiable at some point, say $x(t)=t\sin(1/t)$ at $t=0$, how is its instant $v$ defined? [closed]

If instant velocity at any given time $t_0$ is defined as the derivative of $x(t)$ at $t_0$, what if the derivative does not exist? How are we supposed to deal with $x(t)=|t|$ at $t=0$, or for more ...
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Avoiding a confusion with dot product

Some days ago I have asked a question about a formula for power, many generous people have answered my question and clarify for me that the correct formula of work is $$\mathrm{d}W= \mathbf{F}\cdot \...
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Do partial derivation respect to velocity and total derivation respect to time commute? [duplicate]

Imagine we have a function of position $x^i$ and velocity $v^i$ $f(x,v)$. Position and velocity are both functions of time $t$. If the function doesn't depend explicitely on time, then we have the ...
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Generalization of straight line motion under constant acceleration

My question is that, we all know the three equations of straight line motion under constant acceleration, \begin{align} x & =x_{\rm o}+v_{\rm o}\,t+\tfrac12 \mathrm a\,t^2 \tag{1d-a}\label{1d-a}\\ ...
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1answer
42 views

Defining the exterior derivative with torsion [duplicate]

As introduced in most standard GR textbooks, the exterior derivative of a p-form $X$ is defined in a coordinate basis as $(dX)_{\mu_1....\mu_{{p+1}}}:= (p+1) \partial_{[\mu_1}X_{\mu_2....\mu_{{p+1}]}}$...
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51 views

Expressing acceleration in terms of velocity and derivative of velocity with respect to position

we know that $$a = \dfrac{dv}{dt}$$ dividing numerator and denominator by $dx$, we get $$a=v\dfrac{dv}{dx}$$ provided that $dx$ is not equal to zero or instantaneous velocity not equal to zero when I ...
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Small doubt on derivatives acting on kets/bras

I have a quick, silly question. If $\psi(x):=\langle x|\psi\rangle$, does the bra $\langle x|$ 'go through' the $\partial_x$ operator, as in $$\langle x|\partial_x|\psi\rangle=\partial_x\psi(x) \quad ?...
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I differentiated and drew the graph. It is right? [closed]

Draw the velocity time graph when the displacement of the particle obeys the relation $$s=4+5t+2t^2.$$
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Tensor contraction and covariant derivative

I have naive question about GR and Covariant derivative. You know \begin{align} \nabla_{\gamma} g_{\alpha \beta}=\nabla_{\gamma} g^{\alpha \beta}=0 \end{align} And I would like to compute covariant ...
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81 views

How to relate these two formulas of angular velocity?

Consider the following picture: with $\vec{v}(t)$ the velocity of a particle at time t, $\vec{a}(t)$ its acceleration at time t and $\vec{a_n}(t)$ its normal acceleration at time t. I want to ...
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69 views

What is the meaning of the equation of the change in entropy? [duplicate]

In my chemistry book, the formula for change in entropy is given as : $$\int{dS} = \int{\frac{δq_{rev}}{T}}$$ What is the meaning of $δq_{rev}$? I know that it is the heat exchanged in a reversible ...
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How to derive $\nabla_{\mu} T^{\mu\nu}=\frac{1}{\sqrt {-g}}\partial_{\mu}(\sqrt{-g}\,T^{\mu\nu})$?

This $$\nabla_{\mu} T^{\mu\nu}=\frac{1}{\sqrt {-g}}\partial_{\mu}(\sqrt{-g}\,T^{\mu\nu})\tag{3.39}$$ is from a textbook on general relativity on black hole Vaidya metric, where only non-zero term of ...
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50 views

Classical text of mathematics/infinitesimals for Landau-Lifshitz

I believe their is a pre- and post Weierstrass era of mathematics (loosely speaking). Afterwards there was epsilon-delta, before 'infinitesimals' (with certain rules, ideas and theorems, of course not ...
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68 views

Is motion in infinitesimal interval is linear?

As a kind of thought experiment I tried to think if a motion (including circular motion), when divided into infinitesimal time intervals is always linear motion (whether each interval of the motion is ...
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269 views

Covariant derivative of the spin connection

I wish to compute $$[\nabla_{\mu}, \nabla_{\nu}]e^{\lambda}_{~~a}. $$ To do so, I make use of $\nabla_{\nu}e^{\lambda}_{~~a} = \omega_{a~~~\nu}^{~~b}e^{\lambda}_{~~b}$, so that I may write $$\nabla_{\...
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123 views

Second derivative of a function in a manifold [closed]

Suppose we have a function curve $\gamma(t)$ on a manifold $M$ . Define the function $$f(t)=\frac{1}{2}g_{ab}\dot\gamma(t)^a\dot\gamma(t)^b$$ Introducing coordinates $x^i$ the first derivative of the ...
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53 views

How to express the elementary work definition as an explicit functional expression [duplicate]

My assumption here is that in the definition of elementary work : $dW = F ds$ symbol $d$ represents a differential. But a differential implies a function : $dy =_{df} d[f(x)] = f'(x) \Delta x = f'(...
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Why is $ \text{div}(\vec{r}/r^3) = 0 $?

Let's begin with some context. I was reading Griffiths's introduction to electrodynamics. This makes sense given all the content one can find online in order to understand visually the divergence. ...
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Applying gradient in spherical coordinates to vector in cartesian coordinates [closed]

I am trying to calculate the gradient of a vector field $\boldsymbol{u}$. In cartesian coordinates, I would normally do $$\left(\nabla\boldsymbol{u}\right)_{ij}=\partial_{i}u_{j}=\left(\begin{array}{...
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Spatial derivative of the Hamiltonian Operator

I have a Hamiltonian of a semiclassical wavepacket in some external potential (say an EM field). I would like to linearize this Hamiltonian around a point $x_c$. Intuitively, I would simply try to do ...
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The contravariant derivative of a substitution for the de Sitter metric

Consider the de Sitter metric: $$ds^2 = (1-\frac{r^2}{a^2})dt^2 -(1-\frac{r^2}{a^2})^{-1}dr^2-r^2d\Omega^2$$ I know that we rewrite the metric as $(u,r,\theta \phi)$ using the substitution $$u = t-...
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Where does one more '$\rm m$' come from in the units?

$$\nabla \times A = B$$ $A$ is vector magnetic potential, $\mathrm{Wb/m}$ $B$ is magnetic field intensity, $\mathrm{Wb/m^2}$ Where does one more m come from for $B$? Is that from the gradient operator ...
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87 views

Derivative with respect to vector of a function depending on vectors

I've been trying to understand this concept for hours without any success. I found similar questions on this forum (Derivative with respect to a vector is a gradient?) but I still don't understand. ...
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Given that $m \dot v \cdot v = 0$ , how is it equal to $m \frac{d}{dt} (v \cdot v)/2$? [closed]

While studying about scalar triple product in vector algebra, I stumbled upon the following question with the solution. I want know how is $m \dot v \cdot v $ = $m \frac{d}{dt} (v \cdot v)/2$?
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168 views

Covariant gradient - What am I missing?

I know that the components of the gradient should transform covariantly, and have used this many times in special relativity, etc. However, I also know that covariant components transform like the ...
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55 views

Killing equation in coordinates

In proving that it is possible to write the killing equation in coordinates as $$L_X g=0\iff X_{\alpha;\beta}+X_{\beta;\alpha}=0$$ I have read that the key observation, to write the equation in ...
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Analogous notation to $\nabla$ but for gradient with respect to $\vec{k}$ not $\vec{x}$

$\nabla = \frac{\partial}{\partial x_i}$ so $\nabla F = (\frac{\partial F}{\partial x}, \frac{\partial F}{\partial y}, \frac{\partial F}{\partial z})$. However, is there a similar equalivalent notion ...
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Question about operation with derviative of product in Noether proof

I have been studying Noether's theorem proof, I have a problem understanding one of the last steps, because I don't understand the calculation. Noether theorem assures us that the action under a ...
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Partial derivative notation in thermodynamics

Most thermodynamics textbooks introduce a notation for partial derivatives that seems redundant to students who have already studied multivariable calculus. Moreover, the authors do not dwell on the ...
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467 views

Why is the covariant derivative of the metric tensor with UPPER indices equal to zero? [closed]

I've shown that $\nabla_{\lambda} g_{\mu\nu} = 0 $ rigorously by the following method: $ \nabla_{\lambda} g_{\mu\nu} = \partial_{\lambda}g_{\mu\nu} - \Gamma^{\rho}_{\lambda\mu} g_{\rho\nu} - \Gamma^{\...
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119 views

Different definitions of exterior derivative

In Sean Carroll's GR book, pg 84, the exterior derivative $d$ is defined as $$(dA)_{\mu_1\mu_2...\mu_{p+1}} = (p+1) \partial_{[\mu_1}A_{\mu_2...\mu_{p+1}]}\tag{2.76}$$ where $A$ is a $p$-form and the ...
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In equation (8.37) of the Feynman lectures, it says $H_{ij}$ are derivatives with respect to $t_2$ of the coefficients $U_{ij}$. How do we show this?

In describing how states change with time Feynman first shows that, for small $\Delta t$, each coefficient $U_{ij}$ should differ from $\delta_{ij}$ by amounts proportional to $\Delta t$, like this, $$...
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Using Christoffel symbols to derive formulas for div, grad, curl

In Sean Carroll's GR book, pg. 1oo, it was said that in flat space, the Christoffel symbols vanish in Cartesian coordinates. However, in curvilinear coordinates, they do not vanish. For example, for ...
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128 views

Chain rule for covariant derivative?

Does a chain rule for the covariant derivative exist so that we can evaluate an expression like $$\nabla_c\sqrt{t_{ab}}?$$ where $t_{ab}$ are tensor components? More generally, how do we take the ...
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3answers
193 views

Is the equation $[\nabla_{\mu},\nabla_{\nu}]=F_{\mu\nu}$ correct? If yes, how does it have to be interpreted?

It seems like simply using the equation \begin{equation} \nabla_{\mu}=\partial_{\mu}+A_{\mu} \end{equation} isn't enough: One obtains \begin{equation} [\nabla_{\mu},\nabla_{\nu}]=\underbrace{[\...
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1answer
101 views

Commutator of covariant derivative for rank 2 tensor

I am a newbie at tensor notation and I have been told to prove the identity $$ (\nabla_a\nabla_b - \nabla_b\nabla_a)X^a_{\ \ \ b}=- R^e_{\ \ \ bcd}X^a_{\ \ \ e}+ R^a_{\ \ \ ecd}X^e_{\ \ \ b} $$ I am ...

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