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2
votes
2answers
60 views

Lack of rigour in usual derivation of Work-Energy Theorem

The derivation of the Work-Energy theorem usually goes as follows: You define the work done on a particle under net force $\vec{F}$ as $$W=\int\limits_C \vec{F}\cdot\mathrm{d}\vec{r}$$ And then you ...
2
votes
1answer
180 views

About field gradient

I read the term field gradient in most of the article about magnetic field. I search it online but most of the explanation is about the math. I wonder in physics, what the gradient field really mean? ...
1
vote
0answers
33 views

How to derive $r, \theta, \phi$ for the sperical coordinate gradient? [migrated]

I'm trying to figure out how to get the gradient in spherical coordinates. I'm as far as the author writes in this answer: http://physics.stackexchange.com/a/78514 and I understand how and why to get ...
2
votes
2answers
71 views

Differential operators in curvilinear coordinates

In the appendix A of Griffith's Electrodynamics text, he cites Spivak's Calculus on Manifolds as a reference more a more complete treatment of taking the gradient, curl, divergence, and Laplacian in ...
1
vote
1answer
88 views

Covariant derivative of Levi-Civita tensor [closed]

I'm currently studying Carroll's GR book Spacetime & Geometry, and ran into some trouble understanding the text at page 99 which says: By using metric compatibility $$(\nabla_{\alpha}g)_{\mu ...
5
votes
1answer
168 views

Physical intuition/interpretation of fractional derivatives/integrals?

Oftentimes, when the derivative and integral operations are introduced within the realm of physics, we are taught some physical interpretation of them: Velocity is the derivative of position ...
0
votes
2answers
41 views

Eulerian mass conservation on a stream line to Lagrangian mass conservation

if the density of a fluid particle is conserved on a streamline, $$\frac{d\rho}{dt}=0.$$ Why does this mean $$\frac{\partial \rho}{\partial t}+(\mathbf{v}\cdot\nabla)\rho=0$$ is true everywhere? Why ...
1
vote
0answers
46 views

How is Infinitesimal coordinate transformation related to Lie derivatives?

I am reading the book "Gravitaion and Cosmology" by S. Weinberg. In section 10.9, while discussing Lie derivatives of tensors of different ranks, he makes a general comment: The effect of an ...
1
vote
1answer
40 views

Lagrangian formalism (demonstration)

My question is about the multiplicity of the Lagrangian to a Physics system. I pretend to demonstrate the following proposition: For a system with $n$ degrees of freedom, written by the ...
2
votes
2answers
6k views

Derive vector gradient in spherical coordinates from first principles

Trying to understand where the $\frac{1}{r sin(\theta)}$ and $1/r$ bits come in the definition of gradient. I've derived the spherical unit vectors but now I don't understand how to transform ...
1
vote
1answer
61 views

What is the function type of the generalized momentum?

Let $$L:{\mathbb R}^n\times {\mathbb R}^n\times {\mathbb R}\to {\mathbb R}$$ denote the Lagrangian (it should be differentiable) of a classical system with $n$ spatial coordinates. In the action ...
0
votes
1answer
49 views

Show $\frac{\partial T}{\partial \dot q_j} = m_i \dot r_i^T\frac{\dot r_i }{\partial \dot q_j} $ [closed]

This is a basic result in lagrangian mecanics. Let $T$ be the kinetic energy, $r_i$ be the position of the $i^{th}$ particle in the system I need to show $$\frac{\partial T}{\partial \dot q_j} = ...
2
votes
1answer
52 views

Estimating divergence of set of vectors

I have a set of points where directions and intensities of a flow are given (in 3D). Is it possible to estimate the divergence of the flow defined by those vectors? I only need a rough estimate and I ...
0
votes
1answer
34 views

Meaning the symbol, $W$ and $dW$

What's the difference between $W$ and $dW$? They are both work done and have similar formulae (same dimension). But I don't know the difference between them. $dW$ here ISN'T power.
0
votes
1answer
55 views

Lie derivative in this paper [closed]

In this paper http://arxiv.org/abs/1210.2332 it says in (3.19) p. 8 that $$L_{V}z^A =0$$ but I don't know much about Lie derivatives except what I saw now through wikipedia ...
1
vote
1answer
110 views

Derivative of the magnetic field to the vector potential

So the magnetic field is defined with the vector potential A as: $$\mathbf{B}=\nabla\times\mathbf{A}.$$ How would I calculate the derivative: $$\frac{\delta}{\delta\mathbf{A}}|\mathbf{B}|^2$$ I ...
1
vote
1answer
59 views

Indexed Gradient operator on trigonometric functions

$$\nabla_{i}\nabla_{j}\Big(\frac{\sin(kR)}{R}\Big)$$ Where $R$ is the distance between particle $i,j$. And $k$ is a constant I took $\nabla_{i}=\frac{\partial}{\partial R_{i}}$ and ...
0
votes
1answer
66 views

I need help with divergence and gradient? [closed]

$$A_z = \mu{\frac{e^{-jBr}}{4\pi r}}∫I(z')e^{jBz'\cos\theta}dz'$$ Midway into my question, I want to compute: $$-j\left( \frac{\nabla(\nabla\cdot A) }{w\mu\varepsilon} \right).$$ Symbols like $ w, ...
0
votes
1answer
76 views

Geodesic equation

I have a technical question about the geodesic equation. Assume we have a frame $(E_{1},E_{2},E_{3},E_{4})$ (not necessarily a coordinate frame). Assume we have a parametrized curve $\gamma(s)\in M$ ...
4
votes
2answers
1k views

Derivative of the product of operators and Derivative of exponential

I'm asked to show that $$\frac{d(\hat{A}\hat{B})}{d\lambda} ~=~ \frac{d\hat{A}}{d\lambda}\hat{B} + \hat{A}\frac{d\hat{b}}{d\lambda}$$ With $\lambda$ a continuous parameter. Should I use the ...
2
votes
2answers
57 views

Temperature in statistical mechanics and differentiating entropy

In statistical mechanics, the entropy of an isolated system with energy $E$ (with fixed volume $V$ and chemical composition $N$) is defined as $S(E) = k \log \Omega$, where $\Omega$ is the number of ...
4
votes
1answer
60 views

Computation of $T^{\mu\nu}$ from Lagrangian density $\mathscr{L} $

I am trying to understand how upper and lower indices are connected when computing the energy-momentum tensor. In particular, I found the simple problem where the Lagrangian density is given as ...
2
votes
4answers
1k views

Which Schrödinger equation is correct?

In the coordinate representation, in 1D, the wave function depends on space and time, $\Psi(x,t)$, accordingly the time dependent Schrödinger equation is $$H\Psi(x,t) = ...
0
votes
1answer
65 views

Understanding the covariant derivative and its relation to parallel transport

I have been reading section 3.1 of Wald's GR book in which he introduces the notion of a covariant derivative. As I understand, this is introduced as the (partial) derivative operators $\partial_{a}$ ...
1
vote
0answers
49 views

The Covariant Spinor Derivative in the Locally Supersymmetric Generalisation of the Polyakov Action and Potential Mistakes in the Literature

Questions) I recently came upon the thesis The Landscape of Free Fermionic Gauge Models by D. G. Moore and G.B. Cleaver. On pages 20 and 21 they explain that the locally supersymmetric action, ...
0
votes
2answers
63 views

Divergence of vector potential

I was given the vector potential $$\vec A (\vec r) = - \vec a \times \nabla \frac{1}{r}$$ with a constant vector $\vec a$. Now, I found the $\vec B$ field which is I think $- \vec a \frac{2}{r^3}$, ...
0
votes
0answers
20 views

Bulk Modulus as a function of U and V for fcc lattices

Original bulk modulus equations is $$B=-V\left(\frac{\partial P}{\partial V}\right)\tag{eq 1}$$ At isothermic processes $$P=-\frac{dU}{dV}\tag{eq 2}$$ We can write B in terms of the energy per ...
0
votes
0answers
33 views

Bulk Modulus and its derivative for a fcc lattice

The bulk modulus $B = - V \left(\frac{\partial P}{\partial V}\right)$. At constant temperature the pressure is given by $P= -\frac{\partial U}{\partial V}$, where$ U$ is the total energy. We can ...
0
votes
1answer
26 views

Is difference in wave number always small?

Over the last few days I have been looking at a derivation of group velocity. The derivation is the one shown in this question Deriving group velocity. I have seen this derivation in many places, and ...
1
vote
0answers
100 views

How can I show the time-derivative of an energy functional? [closed]

I have a partial differential equation $$h_t= - \{h^3[I(h)+ \frac {1}{3}h_{xx}]_x\}_x $$ where $I(h)$ is a function of $h$. It results from a fluid dynamics equation. In addition, I have a energy ...
1
vote
0answers
30 views

Lagrangian of a particle on a torus. Calculations right? [closed]

I just want to calculate the motion of a particle on a torus. But it involves some complex calculation. I just want to see if I did everything right. $$f(\phi,\theta)= \begin{pmatrix} (R+ r \cos ...
4
votes
3answers
204 views

Geometric meaning of parallel transport

The definition of parallel transport of a vector $v^b$ along a curve $C$ with tangent field $\it{t}^a$ is given by Wald's GR as $$t^a \nabla_a v^b = 0$$ Is it correct to think of $\nabla_a v^b$ as ...
3
votes
8answers
553 views

Can velocity be an undefined quantity?

We have the image below displaying the uniform velocity by time-distance graph. At every point velocity is constant but what if distance and time both become zero as at origin in the graph is? The ...
1
vote
0answers
22 views

Discrete Laplacian with geodesic distances

Normally, I have a a scalar function f(x,y), sampled on a two dimensional, regularly spaced grid in Cartesian coordinates. Evaluating the Laplacian of this function just requires the standard ...
1
vote
0answers
77 views

Spin connection and covariant derivative

How to prove explicitly (i.e., to don't postulate it) that by including Lorentz indices $a$ the covariant derivative $D_{\mu}$ looks like $$ D_{\mu}A^{\nu a} = \partial_{\mu}A^{\nu a} + ...
1
vote
1answer
128 views

Commutators involving $\Box$ and $\Box^{-1}$ [closed]

How to determine the followings: $$[\Box,\frac{1}{\Box}]\mathcal{O}=?$$ $$[\nabla,\frac{1}{\nabla}]\mathcal{O}=?$$ $$[\nabla^2,\frac{1}{\nabla^2}]\mathcal{O}=?$$ ...
-1
votes
3answers
49 views

Vector question, differentials, Electromagnetism

I was reading this demonstration of electric potential in my book: Let $q$ be a point charge at point $P$ The Electric field created at point $M$ by $q$ is : $$\vec{E}(M) = ...
0
votes
1answer
29 views

Showing $\frac{\delta V_{out}}{V_{out}}=\frac{\delta R_2}{R_2} \frac{R_1}{R_1+R_2}$ [closed]

Consider a voltage divider with $V_{out}=V_{in} \frac{R_2}{R_1+R_2}$. Show that for a small change in $R_2$, the voltage divider equation is: $\frac{\delta V_{out}}{V_{out}}=\frac{\delta R_2}{R_2} ...
1
vote
2answers
79 views

Advection operator

How are exactly $u_j\partial_ju_i$ and $u_i\partial_j u_i$ related? And what is their relation to ($\boldsymbol{u}\cdot\nabla)\boldsymbol{u}$ and $\boldsymbol{u}\cdot(\nabla\boldsymbol{u})$ ? I ask ...
3
votes
2answers
176 views

Trouble with Landau & Lifshitz

Hello I have a quick question on what I have been reading in Landau & Lifshitz's book on classical mechanics. I am in the very beginning of the book and I am having trouble with his derivation on ...
0
votes
1answer
34 views

Maximal Velocity of an Object in Free Fall

A baseball is dropped from a high point. Since the velocity is large, we can say that the drag force is proportional to the square of the velocity, $F_d = \gamma v^2$. My goal is to determine the ...
0
votes
1answer
67 views

Geodesic equation proof confusing me

I was looking through this proof and have no idea where the $u$ comes from. Any help is appreciated. This is from here; I want to know how they got from eqn 5 to eqn 6.
1
vote
2answers
56 views

Can we say that the instantaneous velocity of an object is the displacement in zero time?

Can we say that the instantaneous velocity of an object is the displacement in zero time? In the image above the instantaneous velocity of the object as change in time gets closer and closer to ...
0
votes
0answers
61 views

Are Laplace Operator and mean curvature exactly the same thing for 2D function?

Let's assume we study 2D function/surface f(x,y). Then Laplace Operator is defined as: $$\nabla^2 f=\frac{\partial^2f}{\partial x^2}+\frac{\partial^2f}{\partial y^2}$$ And the mean curvature: let ...
0
votes
1answer
38 views

a mistake related to variable mass system

I'm having a problem with finding my mistake when trying to find the derivative of the momentum when mass is being ejected in a constant rate. The problem is this - a body in space is burning fuel ...
2
votes
1answer
1k views

How is the second-order covariant derivative of a scalar computed?

What is second-order covariant derivative $$\nabla_i\nabla_jf(r)$$ in terms of $r,\theta, g(r)$ and partial derivative, given that the metric takes the form $$ds^2=dr^2+g(r)d\theta^2$$ and $f$ is a ...
6
votes
3answers
1k views

Physics & derivatives written in a weird way

I was always taught that $\frac d {dx} (\ln x) = \frac 1 x$. No derivative had as a result any $dx$ words. In a physics book I encountered something like this (error discussion) [there might be a ...
4
votes
2answers
172 views

Physical meaning of harmonic function?

In complex numbers, we define a harmonic function as a twice continuously differentiable function such that the Laplace operator acting on it gives zero. Can anybody explain me the physical ...
1
vote
4answers
190 views

Rotation systems. Problem interpreting an equation

In this equation: $$ \mathbf a_i\overset{\rm def}{=}\left(\frac{d^2\mathbf r}{dt^2}\right)_i=\left(\frac{d\mathbf ...
19
votes
2answers
748 views

Symbols of derivatives

What is the exact use of the symbols $\partial$, $\delta$ and $\mathrm{d}$ in derivatives in physics? How are they different and when are they used? It would be nice to get that settled once and for ...