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12
votes
2answers
588 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 ...
0
votes
2answers
81 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
28 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
101 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, ...
-2
votes
1answer
70 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
0answers
101 views

$(\frac{\partial S}{\partial P})_T=-(\frac{\partial V}{\partial T})_P$ from $(\frac{\partial S}{\partial P})_V=-(\frac{\partial V}{\partial T})_S$? [closed]

(Source: Nonequilibrium Thermodynamics, 3rd Edition | Yasar Demirel) Background: Nearly all authors just substitute the Maxwell Relation (a) into dH without ever caring about the fact that the kept ...
0
votes
1answer
54 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
69 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}\\ ...
4
votes
2answers
187 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
68 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
131 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 ...
1
vote
1answer
130 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
237 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
135 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?
1
vote
2answers
88 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
35 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 ...
0
votes
0answers
22 views

Where do the minus sign and Laplacian come from in this derivative? [migrated]

I want to find this functional derivative: $$\dfrac{\delta \int d^d x'[\nabla_{x'} \phi(\vec{x}')]^2}{\delta \phi(\vec{x})} = \int d^d x' \left(\dfrac{\delta \nabla_{x'} \phi(\vec{x}')}{\delta ...
2
votes
1answer
57 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
80 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
65 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
697 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 ...
2
votes
2answers
68 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
147 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 ...
2
votes
1answer
62 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
61 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
216 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
2answers
142 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
50 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
93 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
234 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 ...
2
votes
4answers
205 views

Poincare invariant Lagrangians

The Lagrangian density of a Poincare invariant theory should not depend explicitly on the space-time coordinates. Does this mean $$ \partial_\mu \mathcal{L}=0~? $$ If this is the case doesn't the ...
1
vote
2answers
134 views

Time derivative of angular velocity in rotating reference frame

I am going through a section in a textbook regarding the Newton Euler equations for a system of rigid bodies (robotics text). There is a particular line in the derivation I don't understand, I've ...
0
votes
1answer
34 views

The vector r points from $P'(x',y',z')$ to $P(x,y,z)$ [closed]

For some reason this question is giving me a hard time :( The vector $r$ points from $P'(x',y',z')$ to $P(x,y,z)$. (a) Show that if $P$ is fixed and $P'$ is allowed to move, then ...
3
votes
1answer
69 views

Why do we need the material derivative?

I'm studying fluid mechanics, and I got the impression that the material derivative is nothing more than "differentiating along a path" and so I got confused on why do we need it. Basically, let ...
2
votes
1answer
81 views

Relationship between Connection and Material Derivative

Suppose $D\subset \Bbb R^3$ contains a fluid and that $f : D\times \mathbb{R}\to \mathbb{R}$ is a time dependent function defined on the fluid region. In that case, the material derivative is defined ...
0
votes
0answers
46 views

How to calculate electric force between two tubes?

Let's say the electric field due to a charged tube is $E$,length of the charged tube is $l$, radius is $r$ and the surface charge density is $\lambda$. I know that to calculate the electric force ...
0
votes
2answers
96 views

Is there a difference in handwritten nabla $\vec{\nabla}$ with an overset arrow and typeset nabla $\nabla$?

According to some physicist at KIT it is usual to write the following when using pen and paper: whereas in typeset texts you write $\nabla$. Is that true? Are there sources for this convention?
0
votes
4answers
112 views

Dot product of vector and its derivative with respect to time? How does $L \cdot\frac{dL}{dt} = \frac{1}{2}\frac{d(L^2)}{dt}$? [closed]

How does: $$L \cdot\frac{dL}{dt} = \frac{1}{2}\frac{d(L^2)}{dt}$$ where L is a vector (I dunno how to make it bold in the equation). How do they reach to this right hand side equation? And what is ...
0
votes
1answer
25 views

Why does the power in an inductor equal what it does?

I understand that power is that rate at which work is done and that because of this the power in an inductor is equal to $$P=\frac{d}{dt} \left(\frac12Li^2\right).$$ I also understand that the power ...
5
votes
2answers
205 views

A confusion about notation in Goldstein

On treating systems of particles, Goldstein starts with the consideration that whenever there are $k$ particles on a system, the $i$-th one obeys the relation $$\dfrac{d}{dt}{\bf p}_i = {\bf ...
3
votes
1answer
93 views

Why do we do partial and not covariant differentiation with $x^{\nu}$?

Why when taking the velocity vector we make $$u^{\nu}=\frac{d}{d\tau}x^{\nu}$$ and not $$u^{\nu}=\frac{\nabla}{d\tau}x^{\nu}$$ where in the last equation I meant the covariant derivative. Why?
1
vote
1answer
215 views

How to get the time derivative of an expectation value in quantum mechanics?

The textbook computes the time derivative of an expectation value as follows: $$\frac{d}{dt}\langle Q\rangle=\frac{d}{dt}\langle \Psi|\hat Q\Psi\rangle=\langle \frac{\partial\Psi}{\partial t}|\hat ...
1
vote
1answer
57 views

1D Smoluchowski diffusion equation in a linear potential

I am interested in solving a 1D Smoluchowski diffusion equation in a linear potential $U(x) = cx$ for a constant force $c$. This problem follows chapter 4 of the theoretical biophysics script by ...
2
votes
3answers
60 views

Can we measure rates in real time?

I know what it means to say that my position is "X" at a particular moment in time. I can easily take a picture of my motion and observe my exact location at the instant the picture was taken. That is ...
2
votes
2answers
162 views

Any difference between thermodynamic double-derivative and derivative “at constant” value?

Reading about the Maxwell relations has left me confused, and I want a basic sanity check regarding the notation. The Wikipedia article breezes over the following switch of notation without really ...
0
votes
3answers
70 views

Why there is added a partial time derivative in formula for time derivative of potential energy? [duplicate]

In proving the total energy in conservative field is constant we have this equation(picture) why it added partial derivative? Why? I mean where it did come from?
1
vote
0answers
40 views

Partial derivatives in Lagrangian formalism [duplicate]

Suppose I have a function $f = xy$. A partial derivative of $f$ with respect to $x$ implies holding $y$ constant: $$ \frac{\partial f}{\partial x} = y $$ Does this mean that in order to evaluate ...
2
votes
2answers
113 views

Covariant derivative of a covariant tensor wrt superscript

Is it true that when you take the covariant derivative of a covariant tensor, do you always have to do with a subscript? What if you do it wrt a superscript?Does the first term (with the partial ...
4
votes
2answers
132 views

Conventions regarding partial derivatives

Look at this expression: $$\frac{\partial}{\partial t} (V-\mathbf{v}\cdot\mathbf{A}).$$ This expression occurs in Griffiths EM book (4th ed, p.444). $V=V(\mathbf{r},t)$is the scalar potential, ...
1
vote
2answers
67 views

Taylor series: Epsilon not differentiated? [closed]

Why isn't epsilon differentiated with respect to time? (see my question on the right)