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2
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2answers
83 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 ...
1
vote
1answer
95 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 $\partial_{a}$ ...
0
votes
0answers
64 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
118 views

Divergence of vector potential [closed]

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
43 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
80 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 ...
1
vote
1answer
154 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
36 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
49 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 ...
2
votes
1answer
135 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
137 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
52 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
30 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
114 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 ...
0
votes
1answer
38 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
82 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
76 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
82 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
55 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 ...
3
votes
2answers
233 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 ...
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
8answers
613 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 ...
22
votes
2answers
891 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 ...
1
vote
2answers
507 views

How does covariant derivative act on Christoffel Symbols?

the question is how the covariant derivative acts on the following? $\nabla_\nu(\Gamma^\alpha_{\mu\lambda}R^{\beta\lambda})=?$ and ...
2
votes
1answer
53 views

a problem on finding acceleration by differentiation

The displacement of particle along the $x$ and $y$ axis is \begin{cases} x(t)=\omega t-\sin\omega t\\ y(u)=1-\cos\omega t \end{cases} Upon differentiation, the velocity is \begin{cases} ...
3
votes
1answer
158 views

Time derivative of time-translation Killing vector

I'm working with the spherically symmetric, static black hole metric. In the problem I'm working on, I'm told that $K$ is the time-translation Killing vector, $\frac{\partial}{\partial t}$ or $K = (1, ...
-1
votes
1answer
78 views

Finding solution to this differential equation

In this paper http://arxiv.org/abs/hep-th/9506035 equation (3.11) was written as: $$\frac{\partial L}{\partial u}\frac{\partial L}{\partial v} = -1$$ The author then said p.9 that "approximate ...
0
votes
1answer
63 views

Gradient of two-particle system

I'm working on problem 5.1a from Griffiths Intro to QM and given that: $$\mathbf R \equiv \frac{m_1\mathbf{r_1} + m_2 \bf r_2}{m_1+m_2}$$ and $\bf r \equiv \bf r_1 - \bf r_2$ I need to show that, ...
2
votes
2answers
102 views

Derivation of velocities in the Coriolis force

In Fitzpatrick's Newtonian Dynamics book on the Coriolis force, he states \begin{align} v_{x'}&\simeq V_0\cos\theta-2\Omega t V_0\sin\lambda~\sin\theta \tag{433}\\ ...
3
votes
2answers
222 views

Why doesn't this multiplication of Grassmann variables give the expected result?

Would anyone be able to tell me how srednicki goes from step $(44.29)$ to $(44.30)$? Here is the paragraph: Now let us introduce the notion of complex Grassmann variables via $$\begin{align} ...
2
votes
1answer
91 views

Why do derivatives act on vector fields on a worldsheet?

The covariant derivative of a vector $A^{\mu}$ at a point $x$ is defined as $$D_z A^{\mu}=\partial_zA^{\mu}+\Gamma^{\mu}_{\rho\sigma}(x)\partial_{z}x^{\rho}A^{\sigma}$$ where Greek symbols are ...
1
vote
4answers
205 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 ...
2
votes
1answer
264 views

Total and partial derivatives in thermodynamics and Maxwell relations

Consider the expression $$dS=\left(\frac{\partial S}{\partial T}\right)_VdT+\left(\frac{\partial S}{\partial V}\right)_TdV$$ I'm trying to understand how to derive an expression for $\left( ...
1
vote
0answers
241 views

Why discrepancies in the Schrödinger equation? [duplicate]

Why is there seemingly two definitions of the Schrödinger equation? \begin{equation} i\hbar\frac{\partial}{\partial t}\Psi=\hat H\Psi. \end{equation} And \begin{equation} i\hbar ...
1
vote
2answers
167 views

Why is $\nabla\cdot(\hat{\bf r}/r^2)$ giving 0 as answer? [closed]

While I was reading I encountered the statement $\nabla\cdot(\hat{\bf r}/r^2)$ (r cap divided by $r$ square) is 0. Can anyone explain proof of the statement why is it giving 0?
2
votes
2answers
270 views

Derivation of the Riemann tensor confusion

I'm trying to understand the derivation of the Riemann curvature tensor as given in Foster and Nightingale's A Short Course In General Relativity, p. 102. They start by giving the covariant derivative ...
0
votes
1answer
100 views

Want to know about divergence [duplicate]

Can anyone please explain how to know whether a vector field has divergence or not by seeing its diagram? I have read that a vector field must change for having divergence but why is divergence zero ...
2
votes
1answer
74 views

Is this covariant derivative identity true?

Trying to work through a textbook derivation of the geodesic deviation equation, I've calculated this identity:$$u_{;\beta}^{\alpha}u_{\alpha}=u_{\alpha;\beta}u^{\alpha}.$$ If this is true, I'm ...
0
votes
1answer
151 views

Covariant derivative of stress-energy tensor for a scalar field [closed]

In order to prove that $$\nabla ^\mu T_{\mu\nu} =0$$ I want to find the covariant derivative of $$T_{\mu\nu} = \partial_\mu\phi \partial_\nu \phi -\frac{1}{2}g_{\mu\nu}(g ...
2
votes
0answers
80 views

Problems while doing $\dfrac{\partial}{\partial(\partial_\mu \phi)}$ and $\dfrac{\partial}{\partial(\partial_\mu A_\mu)}$

In David Tong's lectures, he gives two Lagrangians as examples to derive the equations of motion: $$\mathcal{L} = \dfrac{1}{2}\eta^{\mu\nu}\,\partial_\mu\phi\,\partial_\nu \phi-\dfrac{1}{2}m^2\phi^2, ...
2
votes
4answers
815 views

Why can't impulse be instantaneous?

We know from 2nd law of motion that $$\vec{F} = \frac{d\vec{p}}{dt}.$$ Now, a rate of change can be instantaneous. So, rate of change of momentum is instantaneous. But I doubt how can there be ...
4
votes
2answers
285 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 ...
3
votes
2answers
1k views

Why and when do we differentiate or integrate equations in physics? [closed]

I'm an engineering student and none of my professors ever explained why do we use derivations and/or integrations in physics. So I have this task, it goes like: The object is moving in a positive ...
1
vote
1answer
170 views

Total vs partial time derivative of action

I'm following Ref. 1 in my reasoning, struggling with action as a function of time. Consider a Lagrangian $$L=\dot x^2-x^2.\tag1$$ Solving the corresponding equations of motion with initial ...
2
votes
2answers
83 views

Determining Acceleration Based On Graph

I understand how to solve this problem, but I am unsure how to generate an equation for the graph (below). My current attempt involves using the mass provided along with the derivative of the line ...
1
vote
5answers
528 views

What is divergence?

What is divergence? I was learning about Maxwells equations and don't understand the divergence part of it. Can someone give an intuition of what divergence is in relation to maxwells equation. To ...
4
votes
3answers
257 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 ...
0
votes
1answer
69 views

Finding the divergence of this force [closed]

I've got to find the divergence of this force, $$ \mathbf F=\left(x^2+y^2+z^2\right)^n\left(x\hat e_x+y\hat e_y+z\hat e_z\right) $$ I would know how to do it if the $n$ superscript wasn't there. Any ...
2
votes
1answer
95 views

Can these two terms cancel out?

In trying to prove that $$\Gamma_{\mu\nu\lambda} = \eta_{ab}J^a_bJ^b_{\nu\lambda}.$$ The author canceled out while expanding the first equation $$J^a_{\mu\lambda}J^b_\nu$$ with ...
1
vote
2answers
256 views

Differentiation in general relativity

If we have: $$ \frac{d\phi^a}{d\tau}= \frac{\partial \phi^a}{\partial x^\mu} \frac{dx^\mu}{d\tau} \tag{1}$$ Differentiating it, we get: $$ \frac{\partial \phi^a}{\partial ...