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5
votes
2answers
116 views

Intuitive analysis of gradient, divergence, curl

I have read the most basic and important parts of vector calculus are gradient, divergence and curl. These three things are too important to analyse a vector field and I have gone through the physical ...
0
votes
0answers
23 views

Relation between differentiation of one-form basis and Christoffel Symbols

If I want to covariantly differentiate a one form then I can write: $\nabla_\beta \tilde p = \dfrac{\partial p_\alpha}{\partial x^\beta} \tilde \omega^\alpha + p_\alpha \dfrac{\partial \tilde \omega^...
1
vote
1answer
129 views

Starting from an expression of E(V) and P(V) for the Birch-Murnaghan's equation of state, is there a way of obtaining an expression for E(P)?

I have a function, an equation of state named the Birch-Murnaghan's equation of state, in which the Energy ( $E$ ) is a function of the Volume ($V$), where $E_{0}$, $V_{0}$, $B_{0}$ and $B_{0}^{'}$ ...
0
votes
0answers
38 views

differentials in physics

Often I find the following expressions in physics books: Say we have a current density $\vec{j}=\rho\vec{v}$ through a surface $\vec{F}$ of particles $N$ in the volume $V$ with the density $\rho=dN/dV$...
0
votes
1answer
34 views

How to treat the units of measure when taking a derivative?

I've had a doubt for a long time: when I'm taking the derivative, of a function for example, how should I treat the units of measurement? For example, if I'm taking the derivative of: $$S\,[{\rm m}]=...
6
votes
3answers
90 views

Vlasov equation, Maxwell distribution

I have the Maxwellian distribution: $f(v)=n(\frac{m}{2\pi kT})^{\frac{3}{2}}\exp\left(-\frac{mv^2}{2kT}\right)$ I have to show that it is a solution to the Vlasov equation: $\frac{\partial f}{\...
1
vote
4answers
49 views

Area under and slope of the motion graphs

I wanted to ask in general what area under the graph means. Also which physical quantity is highlighted by area under distance vs time graph. I'm confused that area is a 2 dimensional concept and it ...
-1
votes
1answer
62 views

Covariant derivative [closed]

Hi, Could you explain to me why the subtraction of vector at some point and parallel transported vector is covariant derivative vector. How is it possible
-1
votes
1answer
53 views

Confusion in differentiation in physics problem [closed]

Here, we had to find theta such that the denominator has the maximum value. Being new to differentiation I basically didn't understand how differentiation solved the purpose: I basically didnt ...
0
votes
0answers
45 views

Difference between integral and differential physical laws [duplicate]

Why is integral and differential physical laws both used? I read that integral is global and differential is local. Could you tell me something about it?
0
votes
1answer
65 views

Use of infinitesimals in physics [duplicate]

I want to ask about infinitesimals and non-standard analysis. In physics we always use $\mathrm dx,~\mathrm dv,~\mathrm dt$ etc. as infinitesimal quantities. When we deduce equations in physics, when ...
0
votes
0answers
44 views

Help with understanding what is Curl [duplicate]

Yeah, I watched several YT videos and read few articles and my head is spinning. I am trying to get the right understanding of what Curl is. There is this excellent video: Divergence and Curl Now ...
0
votes
1answer
69 views

Yang-Mills field strength tensor

In basically every QFT book the Yang-Mills strength tensor $F_{\mu\nu}$ is defined as $$F_{\mu\nu}=[D_\mu,D_\nu]$$ where $D_\mu$ is the covariant derivative $$D_\mu=\partial_\mu-A_\mu$$ and $A_\mu$ is ...
1
vote
1answer
37 views

Second covariant derivative, computation problem

I am having a question on the wikipedia article http://en.wikipedia.org/wiki/Second_covariant_derivative Using the notation therein I don't get why $(\nabla_{u}\nabla_{v}w )^a=u^c\nabla_{c}v^b\nabla_{...
1
vote
3answers
91 views

Covariant Derivative of Kronecker Delta

I am reading Carroll's book on GR right now, and I ran into a little trouble in his chapter 3 on curvature. He is establishing the properties of the covariant derivative, and claims that the fact that ...
0
votes
1answer
27 views

Equations of motion acceleration doubt

So i was going through some text today morning. Where it said $$ a = \frac{vdv}{dx} $$ So they then went on to, $$ vdv = adx \\ \implies \int vdv = \int adx$$ But,I am very certain acceleration is ...
0
votes
0answers
23 views

Partial Differentiation without chain rule in Euler Lagrange Equations [duplicate]

The Euler-Lagrange equations for a bob attached to a spring are $$ \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial v} = \frac{\partial L}{\partial x} $$ But $v$ is a function of $x$. Is it ...
0
votes
1answer
32 views

Commuting of the covariant derivative: Menzel's Mathematical Physics

Menzel defines covariant differentiation as equivalent to partial differentiation with respect to the general coordinates. “To indicate the covariant nature of the differential operator, set $$\frac{\...
0
votes
1answer
41 views

Derivation of centripetal acceleration

While reading HC Verma chapter 7 circular motion I came across a derivation which I couldnt understand. I have marked my doubt with red. I don't understand from where +dw/dt [- i sine +j cos0] came ...
-1
votes
1answer
41 views

Maximum electric field of a circular ring

How do you differentiate the equation for electric field of uniform ring $$ E_x = \frac{kxQ}{(x^2+r^2)^{3/2}} $$to get the maximum at a point? My book says $x = \frac{r}{\sqrt2}$. I tried ...
4
votes
1answer
112 views

Covariant derivative of a covariant derivative

I'm trying to find the covariant derivative of a covariant derivative, i.e. $\nabla_a (\nabla_b V_c)$. This is something I've taken for granted a lot in calculations, namely I though that by the ...
4
votes
3answers
216 views

Is $\dfrac{dx}{dt}$ a fraction or not?

I am new to calculus and during my mathematics class my sir defined $\dfrac{dx}{dt}$ as $$dx/dt=\lim_{t\to t_1}\dfrac{f(t)-f(t_1)}{t-t_1}$$ and my sir made a clear statement that $\dfrac{dx}{dt}$ ...
0
votes
1answer
50 views

Vector Derivative: General Case

From "An Introduction to Mechanics" by Kleppner & Kolenkow, SIE-2007, Chapter 1 (Vectors and Kinematics), Section 1.8 - "More about the derivative of a vector". In this section, towards the ...
0
votes
1answer
37 views

Differential Operator

I am trying to understand the following expression \begin{eqnarray} e^{-ik.x}D_{\mu}D^{\mu}e^{ik.x} & = & e^{-ik.x}(i\partial_{\mu}+A_{\mu})(i\partial^{\mu}+A^{\mu})e^{ik.x}\\ & = & e^{...
3
votes
2answers
45 views

Variation of Lagrangian density $\mathcal{L}$ w.r.t $x_\mu$

If a function $f(x(t),y(t))$ has no explicit dependence on the variable $t$, then $\frac{\partial f}{\partial t}=0$. In quantum field theory, the Lagrangian density $\mathcal{L}(\phi,\partial_\mu\phi)...
-3
votes
1answer
45 views

Lagrangian in polar coordinates [closed]

$$L=\frac{1}{2}mv^2=\frac{1}{2}m(\dot{x}^2+\dot{y}^2)$$ $$L=\frac{1}{2}mv^2=\frac{1}{2}m(\dot{r}^2+r^2\dot{φ}^2)$$ I dont get this part. $$\frac{d}{dt}\left(\frac{\partial{L}}{\partial{\dot{φ}}}\...
3
votes
2answers
53 views

How to derive wave speed/tension relation for the vibrating string?

I was studying vibrating strings and in my teacher's notes I found that, generically, if I change the tension on the string by $\Delta T$ then, the speed percentage change can be written as: $\frac{\...
1
vote
0answers
21 views

Position, velocity, acceleration, jolt, and [duplicate]

I am familiar with the fact that $\displaystyle{\frac{dx}{dt}}=v$, $\displaystyle{\frac{dv}{dt} =a}$, and $\displaystyle{\frac{da}{dt}=J}$ where $J$ denotes the 'jolt', or jerk. Are further ...
0
votes
0answers
41 views

Different subscripts for $\nabla$ operators while deriving force on system of many particles

Consider a system of 4 particles in an external conservative field. So force acting on each particle is derived from potential energy $U(x,y,z)$of the particle+field system: Total (external) force on ...
3
votes
1answer
44 views

Derivative with respect to a difference of independent variables

I am dealing with an equation from nonlinear acoustics (Khokhlova-Zabolotskaya-Kuznetsov equation) where a strange term (for me as a mathematician) is used. The equation looks like this $$ \frac{\...
3
votes
1answer
52 views

Is this equation $\nabla_a\sqrt{-g}=0$ correct? [duplicate]

Is the equation $$\nabla_a\sqrt{-g}=0$$ correct? Here $\nabla_a$ is the Levi-Civita connection, and $g$ is the determinant of metric $g_{ab}$. Apparently, we have $\nabla_ag_{bc}=0$, but I am not sure ...
4
votes
2answers
146 views

Why is the covariant derivative of the determinant of the metric zero?

This question, metric determinant and its partial and covariant derivative, seems to indicate $$\nabla_a \sqrt{g}=0.$$ Why is this the case? I've always learned that $$\nabla_a f= \partial_a f,$$ ...
1
vote
0answers
76 views

Euler-Lagrange equations in General Relativity

When obtaining the Euler Lagrange equations for a scalar field with higher order derivatives in curved space is it the same to use $$ -\partial_\nu\partial_\mu\frac{\partial \sqrt{-g} \mathcal{L}}{\...
1
vote
0answers
38 views

Are physical functions always differentiable [duplicate]

I know that physicist usually don't really think too much about differentiabillity of functions. Usually there are at most finite many points where functions aren't differentiable and if there are ...
0
votes
1answer
56 views

Can there be a wave function that is physically possible but is non differerentiable (maybe even non-continous)?

The definition of a wave function demands continuity and differentiability so that it can satisfy the Schrödinger Equation. My question is whether this assumption is necessary for reality. Does ...
1
vote
1answer
38 views

Quick question - infinitesimals proofs [duplicate]

In a few of my courses in mechanics certain statements/equations have been proved by assuming that two infinitesimals multiplied by each other are zero. For instance in the equation : $dx + dy + dx^2 ...
0
votes
1answer
35 views

Directional derivative of the potential energy in the direction of the displacement in three dimensions

For a conservative force $\vec{F}=-\vec{\nabla } U \implies \mathrm dW= -\vec{\nabla} U \cdot \mathrm d\vec{s} $ Where $\mathrm d\vec{s}$ is the infinitesimal displacement. For a differentiable ...
1
vote
2answers
58 views

Infinitesimal time intervals use

I've a question, that maybe will sound obvious, on the use of infinitesimal quantities. Consider the expression for the acceleration in non inertial frames. $\frac{d\vec{v}}{dt}=\frac{d\vec{v'}}{dt}...
0
votes
2answers
59 views

Calculate divergence via partial derivative [closed]

I need to calculate the divergence and curl for a vectorfield. I've done that before so that's no problem :) Or I've done it using partial derivative, maybe there are multiple ways to solve for ...
0
votes
1answer
48 views

Help with relativistic notation (Derivative of Lagrangian)

I am trying to learn QFT, but I haven't taken a course in general relativity so the relativistic notation stuff is taking me a bit to get used to. I do not understand how to do the following. For a ...
-1
votes
1answer
51 views

Uncertainty in Range of Projectile [closed]

If we are given that a projectile is launched with velocity 10m/s at an angle of $45^\circ$ and uncertainty in angle is of $0.5^\circ$ . What is the uncertainty in the range of projectile. The problem ...
2
votes
2answers
83 views

Do integrals of position make any sense? Do they have an application? [closed]

I know that taking the derivative of position with respect to time defines what we call velocity, but I've never heard of physicist going in the opposite direction with position. Is there any ...
1
vote
1answer
57 views

Killing equation manipulation

Why does the killing equation $$K_{\mu;\nu}+ K_{\nu;\mu} = 0$$ equal $$K_{\mu,\nu}+ K_{\nu,\mu} -2\Gamma^{\rho}_{\mu\nu}K_{\rho} = 0 $$ when in general a covariant derivative $V_{\beta;\alpha} = (\...
1
vote
2answers
67 views

Free falling and bouncing back

My confusion arises with free falling body. For a free falling body the displacement ~ time graph has a kink (at the time when the body hit the ground ). at a kink point, a function is not ...
0
votes
1answer
34 views

Calculating motion of equation in tensor form

for the Lagrangian density $$\mathscr{L}=\frac{1}{2}(\partial_{\mu}A^{\mu})^2$$ how can I get this $$\frac{\partial{\mathscr{L}}}{\partial(\partial_{\mu}A_\nu)}=(\partial_\rho A^\rho)\eta^{\mu\nu}$$ ...
1
vote
0answers
57 views

Partial derivative vs Total derivative

This is essentially a follow up to my question here since I seem to have some difficulties regarding the differences between partial and total derivatives. Consider a Lagrangian density $$\mathcal{...
1
vote
1answer
38 views

Integral of a velocity profile?

Part of my fluid mechanics homework asks me to solve: $${\partial u\over \partial x} = 0$$ Which represents how the velocity profile, u, changes in the x. I'm not sure whether you can integrate ...
1
vote
1answer
52 views

Meaning of dt/dx when deriving the law of reflection

One way to derive the law of reflection, you can use the principle of least action to minimize the action path of motion of light. They key concept while doing this is to take the derivative of the ...
0
votes
0answers
46 views

Confused about something in Landau classical mechanics

On page 7 in Landau and Lifshitz Mechanics He writes: We have $L'=L(v'^2)=L(v^2+2ve+e^2)$ now the confusing part comes (for me): He writes: Expanding this expression in powers of e and neglecting ...
1
vote
0answers
61 views

Why is this differential added instead of subtracted?

I was looking at a derivation of the Barometric formula which reads like this: Consider a flat disc of air of mass $\mathrm{d}m$ at distance $h$ above the ground of mass $\mathrm{d}m$ and ...