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0
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1answer
31 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
0answers
62 views

What does $dx/dt$ mean in this formula? [on hold]

The formula i'm struggling with is A1a in the appendix of this paper (A double pendulum model of tennis strokes. Rod Cross, University of Sydney. 2006) - LINK. I understand all the other terms of the ...
-2
votes
0answers
16 views

the motion of a body in a resisting medium [closed]

the motion of a body falling from rest in resisting medium is described by the equation dv/dt = (a - bv); where a and b are constants. the velocity at any time 't' is? PS: please explain it fully
-1
votes
1answer
37 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
96 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
199 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
46 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
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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}\\ & = & ...
3
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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 ...
-4
votes
0answers
74 views

${\partial\over{\partial x_j}}\left(\partial x_i\over\partial t\right)\ne{\partial\over{\partial t}}\left(\partial x_i\over\partial x_j\right)$? [migrated]

$ \boldsymbol x = f(\boldsymbol X,t)$ is the position of a particle in an instant of time $\boldsymbol X$ is the initial position $t$ time $\boldsymbol u$ velocity In my opnion $f$ is continuos... ...
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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. ...
3
votes
2answers
47 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: ...
1
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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
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0answers
39 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 $$ ...
3
votes
1answer
50 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
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2answers
125 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 $$\nabla_a f= \partial_a f,$$ ...
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0answers
74 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} ...
1
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0answers
37 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
54 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
34 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
32 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
41 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. ...
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
49 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
55 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
65 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
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0answers
51 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 ...
1
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1answer
37 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
50 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
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0answers
44 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
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0answers
60 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 ...
0
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0answers
31 views

What is the derivative of the length vector in the direction of the original unit vector?

What is the derivative of the length vector? I am asking this question becuase I have seen, in Prof. M. S. Sivakumar's youtube video, the result that $\dot{|r|} \hat{r} = \dot{r}$ Where, $ ...
0
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0answers
34 views

Advection Operator shift in scalar product

Can someone help me with advection operator shifts? I can't figure out the rule for the shift inside of a scalar product. The terms $(u,(v\cdot \nabla)\delta v)_\Omega$ and $(u,(\delta v\cdot ...
3
votes
1answer
66 views

Partial derivatives vs total derivatives in thermodynamics

The specific heat of a system is defined as $$C_z = T \left( \frac{\partial S}{\partial T} \right)_{z=\text{const}}$$ Sometimes however, I find the same definition, but with total derivatives ...
1
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0answers
47 views

How does the gradient operator pick up a minus sign when the reference frame is switched from one particle to another? [closed]

A potential between two particles, $i$ and $j$, is given as a function only of the separation distance, $$V_{ij} = V_{ij}(|r_i − r_j|)$$ It should follow that the force by $j$ on $i$ is equal and ...
0
votes
1answer
65 views

Divergence of inverse of metric tensor

I know that the Levi-civita connection preserves the metric tensor. Is the divergence of the inverse of metric tensor zero, too?! I'm not so familiar with the divergence of the second ranked tensor. ...
3
votes
1answer
26 views

Dimensional interpretation of inverse gradient length $\frac{d}{dx} \ln(Y)$

Preliminary definition: inverse gradient length Let me first explain what I mean by that term. The inverse gradient length of some quantity $Y$ (often thermodynamic temperature $T$) $L_Y^{-1}$ is ...
3
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2answers
94 views

Exact differentials and state functions

I was reading a Wiki article on the relationships between heat capacities And during the derivation I came across this formula (and others like it): This equation was used as a tool in a ...
1
vote
0answers
44 views

Higher order versions of momentum? Can conservation principles be established and used? [closed]

Question Can higher order derivatives of momentum be useful in creating theories of dynamics if they have conservation principles? Even if they aren't needed, could it be done in theory? For ...
0
votes
1answer
72 views

A question when using $E= - \nabla V$

This problem is from Problems and Solutions on Electromagnetism. A thin but very massive disc of insulator has surface charge density $\sigma$ and radius $R$. A point charge $+Q$ is on the axis of ...
1
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3answers
241 views

Time derivative of a function in Phase Space

Consider a function $\mathcal{H}(q_i,p_i;t)$ such that it obeys the equation: $$ \frac{d\mathcal{H}}{dt}=\frac{\partial\mathcal{H}}{\partial t}$$ What does this equation imply (read: mean), ...
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0answers
48 views

Mode expansions of fields

This is a very simple question but would appreciate it if someone could clarify - I've heard different things from different people so I'm a little bit confused yet the question is simple: Given the ...
2
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0answers
98 views

Supercovariant Derivative action

My query is with Weinberg Vol3 equation just above 26.7.22 Weinberg follows Majorana Superfield formalism. Where, covariant derivative is defined as, $$D_{R\alpha}=-\epsilon_{\alpha ...
0
votes
1answer
67 views

Variation of a tensor

Let a change of coordinates be given by $x^{\mu}\to x^{\mu '}=x^{\mu}+\varepsilon \xi^{\mu}(x)$ with epsilon a small quantity. Given a tensor $T$ we define $\delta T:=T'(x)-T(x)$. I guess this means ...
1
vote
1answer
86 views

Derivation Of Euler-Lagrange Equation [closed]

I want the proof of this relation in details, $$ \frac{\rm d}{{\rm d}t}\left(\frac{\partial\vec{r}_v}{\partial q_\alpha}\right)=\frac{\partial\vec{\dot{r}_v}}{\partial q_\alpha} $$
1
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3answers
176 views

What is the physical meaning of the Levi-Civita connection?

I'm taking a course in General Relativity and I have studied the fundamental theorem of Riemannian geometry: Let $M$ be a manifold with metric $g$. Then exists an unique torsion-free connection ...