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3
votes
0answers
39 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 ...
2
votes
1answer
42 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 ...
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 = \...
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
128 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
131 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
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
38 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
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{\...
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 ...
-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
55 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
67 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
72 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
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
96 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
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{\...
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 ...
-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
119 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
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
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{\...
2
votes
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
votes
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 ...
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{\...
3
votes
1answer
54 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
154 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
77 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
39 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
36 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
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
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 ...