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0
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0answers
16 views

How to explain implicit time dependence to someone?

I am trying to explain what implicit time dependence is and how it differs from explicit time dependence, but I'm unsure how "sound" my explanation is. Here is what I said: Suppose I have a function $...
2
votes
2answers
277 views

How to find Tangential/Radial/Angular Velocity for motion in any curve?

Is the radial velocity responsible only for changing distance between objects and the component perpendicular to it only for change in direction? If so why? Please try to give a different explanation ...
3
votes
0answers
40 views

Why do we assume electromagnetic fields to be doubly differentiable? [duplicate]

It seems like the identities of curl of gradient, divergence of curl, and the simple derivations of electromagnetic waves from Maxwell equations all rely on the symmetry (interchangeability of their ...
6
votes
3answers
110 views

Vlasov equation, Maxwell distribution

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

Meaning of $\nabla_{\mathbf{p}_k} W(\mathbf p, h)$ in PBF

I'm reading this paper on Position Based Fluids and I couldn't understand the meaning of $\nabla_{\mathbf{p}_k} W(\mathbf{ p_i - p_j}, h)$ in the equation 7 (see below). …the gradient of the ...
17
votes
2answers
8k views

Difference between $\Delta$, $d$ and $\delta$

I have read the thread regarding 'the difference between the operators between $\delta$ and $d$', but it does not answer my question. I am confused about the notation for change in Physics. In ...
0
votes
1answer
65 views

Could you give me an application on physics of Gauss Divergence Theorem for scalar? [closed]

Gauss divergence theorem for vectors can be easily explained by mass balance. But I can't think about one example for scalar gauss divergence theorem. Gauss Divergence Theorem for scalars: $$\int\...
-1
votes
1answer
59 views

Thermodynamics - Partial Derivatives [closed]

I just need help to solve a problem: $\left(\frac{∂\overline{E}}{∂V}\right)_{β,N} + β\left(\frac{∂\overline{p}}{∂β}\right)_{N,V} = - \overline{p}$ PS: The bar over E and over p (this in both sides) ...
1
vote
2answers
54 views

If change in position over time is average velocity, why doesn't change in position over time squared equal average acceleration?

For example, let's say a car is experiencing an acceleration of $1$m/s$^2$, for $6$ seconds so it goes $18$m. Now the average velocity is found through dividing $18$m by $6$s which is in line with the ...
-4
votes
0answers
34 views

Modulus in Calculus? [migrated]

Does Modulus function has any effect during differentiation and integration of a quantity? For example: Let two velocities be: $$ v_1= (t-2) m/s $$ and $$ v_2=|t-2|m/s $$ If we differentiate them ...
0
votes
0answers
14 views

Why does the material derivative and transport theorem look different?

Reynolds transport theorem says that $ \frac{d\int\phi}{dt}=\int\left(\frac{\partial\phi}{\partial t} + \nabla\cdot(\phi\otimes v) \right) $ Why is the material derivative not defined as what's ...
1
vote
0answers
38 views

Acting for a covariant derivative on charged spinor [closed]

For field, theory what i know $i.e$,complex scalar QED \begin{align} D_\mu \phi = \partial_{\mu} \phi - i Q A_{\mu} \phi \end{align} and \begin{align} D_\mu \phi^{\dagger} = \partial_{\mu} \phi^{\...
0
votes
1answer
36 views

Interpretation of the operation $v^\alpha \nabla _\alpha v^\mu$

In general relativity, we can write the geodesic equation as a contraction $v^\alpha \nabla _\alpha v^\mu = f(\lambda)v^\mu$ along a path defined by coordinates $x^\mu(\lambda)$, and where $v^\mu = \...
2
votes
1answer
639 views

The role of the affine connection the geodesic equation

I apologise in advance that my knowledge of differential geometry and GR is very limited. In general relativity the equation of motion for a particle moving only under the influence of gravity is ...
1
vote
0answers
11 views

Proof from Calculus 1 [migrated]

Last days, from going into a website of the university of Pisa, I found an exercise given in the previous exams, in 1999. The problem was like: Given a continuous function f in R, and which ...
5
votes
2answers
131 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
25 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
132 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 $B_{0}^{'}$ ...
1
vote
2answers
60 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
0answers
41 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
39 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}]=...
1
vote
4answers
56 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 ...
3
votes
2answers
265 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 ...
-1
votes
1answer
67 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
56 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
68 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 ...
1
vote
4answers
189 views

What is the physical significance of curl $\nabla\times\boldsymbol{V}$?

What is the physical significance of curl $$\nabla\times\boldsymbol{V}~?$$ I mean I read 'curl V represents the rotation of the vector $V$. My question what is it about the term $\nabla\times\...
0
votes
1answer
73 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 ...
0
votes
1answer
33 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{\...
1
vote
1answer
41 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
106 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
28 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
44 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 ...
3
votes
1answer
45 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{\...
4
votes
1answer
120 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 ...
-1
votes
1answer
42 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
3answers
218 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
53 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
46 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
54 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{\...
6
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7answers
1k views

Physical intuition for higher order derivatives

Could somebody give me an intuitive physical interpretation of higher order derivatives (from 2 and so on), that is not related to position - velocity - acceleration - jerk - etc?
2
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0answers
22 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
43 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 ...
17
votes
2answers
577 views

Lagrangian Mechanics - Commutativity Rule $\frac{d}{dt}\delta q=\delta \frac{dq}{dt} $

I am reading about Lagrangian mechanics. At some point the difference between the temporal derivative of a variation and variation of the temporal derivative is discussed. The fact that the two are ...
3
votes
1answer
55 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 ...