Questions tagged [differentiation]

Differentiation is the set of techniques and results from Differential Calculus, concerning the calculation of derivatives of functions or distributions.

Filter by
Sorted by
Tagged with
1
vote
2answers
38 views

Derivative of rotation matrix in a form skew-symmetric matrix

I am working on an application of CV, in which a way to calculate the derivative of rotation matrix is involved. $$R$$ is the rotation matrix and $$R \in SO(3)$$ Also, $R$ is changing with $t$ giving $...
0
votes
1answer
37 views

Differentiation/Integration

I am given $v=6-4x$ where $v$ is velocity and $x$ is position of the body. I am now asked to find displacement between $t=1.4$, distance covered from $t=1.4$, average velocity from $t=2.5$, average ...
0
votes
1answer
23 views

What does the derivative of unit vector of velocity with respect to time represent?

let an object move with a constant accelration a. in my book,the following derivatve is said to be non-constant(variable). $$\frac{d[\frac{v}{|v|}]}{dt}$$ what does this mean? as far as i can think,it ...
1
vote
1answer
47 views

Confusion in derivation of Formula of divergence

This is from the book Mathematical Methods (Arfken), Can someone explain how did one del(x) and one dx came here,?
0
votes
1answer
46 views

Which is the differential $\text{d} p_i$ of a generalized momentum?

I want to get a partition function, but I introduce a generalized momentum, my doubt is about, when I define a differential respect to $p$, it means $\text{d} p$, which is the correct form to get it? ...
2
votes
1answer
51 views

Factor before Dirac delta in magnetic dipole field formula

I bumped into this formula for the magnetic induction field generated by a dipole, containing Dirac's delta, while studying hyperfine splitting: $$\textbf{B}(\textbf{r}) = \frac{2}{3}\mu_0 \textbf{m}\...
1
vote
1answer
21 views

Time derivatives of the unit vectors in cylindrical and spherical

In cylindrical and spherical coordinates, the position vectors are given by $\mathbf{r}=\rho \widehat{\boldsymbol{\rho}}+z \hat{\mathbf{k}}$ and $\mathbf{r}=r \hat{\mathbf{r}}$, next to next, and ...
0
votes
2answers
58 views

Spin coherent state path integral derivation

I'm trying to follow the exposition of spin coherent state path integral presented in Condensed Matter Field Theory by Altland and Simons (section 3.3, Page 134-142), and I have a problem with the ...
0
votes
1answer
27 views

Help decipher the notation said to denote a common pattern in various branches of science in Prelude to Mathematics by W. W. Sawyer

In Section 1.2 - Nature's Favorite Pattern? (excerpted below) of Prelude to Mathematics by W. W. Sawyer (1982), he said mathematicians used the notation $\nabla^2 V$ to denote a pattern that occurs &...
2
votes
2answers
82 views

Why is 4-velocity not defined as the covariant derivative of position instead of the regular time derivative?

The geodesic equation is usually written as \begin{equation} D_\tau u^\mu = 0 \end{equation} where $D_\tau= u^\mu \nabla_\mu$ is the covariant proper time derivative and $u^\mu=\frac{dx^\mu}{d\tau}$ ...
2
votes
0answers
43 views

Proving that a map is differentiable

Let $H$ be a self-adjoint operator on $\mathcal{H}$, $\psi\in D(H)$ and $\beta\geq 1/2$. How can I see that $$ L \colon \mathbb{R}\to \mathcal{L}(D(\mathcal{N^\beta}),D(\mathcal{N^{\beta-1/2}})), \...
0
votes
0answers
28 views

Gondolo-Gelmini Change of Variables

In the article Cosmic abundances of stable particles: Improved analysis, P. Gondolo and G. Gelmini, Nucl. Phys. B 360 (1991), p. 145-179, they convert $\rm{d}^3p_1\rm{d}^3p_2=2\pi^2p_1p_2\rm{d}E_1\rm{...
1
vote
1answer
30 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 ...
1
vote
1answer
61 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. (...
0
votes
2answers
39 views

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 ...
0
votes
2answers
59 views

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 \...
2
votes
2answers
58 views

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 ...
4
votes
2answers
85 views

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 ...
0
votes
0answers
20 views

Expressing this result in different coordinates [migrated]

Is there a neat way to express this in Cylindrical and Spherical coordinate systems? $(\vec{A}.\nabla)\vec{B}$ Reference: this occurs quite frequently in Electrodynamics books including Griffiths. ...
0
votes
2answers
108 views

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 - ...
0
votes
1answer
27 views

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 ...
1
vote
2answers
69 views

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\...
1
vote
3answers
95 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) \\ ...
1
vote
0answers
35 views

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 ...
8
votes
2answers
365 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 $\...
0
votes
1answer
41 views

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 $...
1
vote
2answers
63 views

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). ...
2
votes
3answers
104 views

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 ...
-1
votes
3answers
87 views

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 \...
1
vote
0answers
19 views

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 ...
3
votes
2answers
156 views

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}\\ ...
1
vote
1answer
36 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 $(dX)_{\mu_1....\mu_{{p+1}}}:= (p+1) \partial_{[\mu_1}X_{\mu_2....\mu_{{p+1}]}}$...
1
vote
1answer
46 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 ...
2
votes
3answers
90 views

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 ?...
-1
votes
1answer
48 views

I differentiated and drew the graph. It is right? [closed]

Draw the velocity time graph when the displacement of the particle obeys the relation $$s=4+5t+2t^2.$$
0
votes
1answer
69 views

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 ...
0
votes
2answers
74 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 ...
3
votes
2answers
65 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 ...
4
votes
1answer
71 views

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 ...
0
votes
0answers
48 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 ...
2
votes
3answers
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 ...
8
votes
2answers
254 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_{\...
1
vote
1answer
107 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 ...
2
votes
1answer
49 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'(...
3
votes
3answers
143 views

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. ...
-1
votes
1answer
62 views

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}{...
0
votes
1answer
52 views

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 ...
0
votes
1answer
42 views

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-...
4
votes
2answers
50 views

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 ...
0
votes
3answers
82 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. ...

1
2 3 4 5
26