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0
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0answers
30 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
75 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
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4answers
88 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
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1answer
24 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
180 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 ...
1
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0answers
18 views

Normalising an equation, help with differentials [migrated]

So I am trying to normalise equations of the form, $$dy/dx=B \mbox{ and } d^{2}y/dx^{2}=A$$ If I define $y^{*}$ as; $$y^{*}=By \Rightarrow dy^{*}/dy=B $$ Is it also then true that, $$d(dy^{*})/dy = B ...
3
votes
1answer
80 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
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1answer
61 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
24 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
51 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
120 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
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3answers
49 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
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0answers
30 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
81 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
127 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
51 views

Taylor series: Epsilon not differentiated? [closed]

Why isn't epsilon differentiated with respect to time? (see my question on the right)
1
vote
3answers
186 views

Apparent dimensional mismatch after taking derivative

Suppose I have a variable $x$ and a constant $a$, each having the dimension of length. That is $[x]=[a]=[L]$ where square brackets denote the dimension of the physical quantity contained within them. ...
1
vote
1answer
64 views

Curl of a vector field with two different systems of coordinates

Let $$\mathbf{H} = H_x \mathbf{u}_x + H_y \mathbf{u}_y + H_z \mathbf{u}_z$$ be a vector field whose components are defined with respect to the unit vectors $\mathbf{u}_x$, $\mathbf{u}_y$ and ...
1
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2answers
50 views

Trouble with derivation in an equation for Newton's Law of Angular Motion

I'm an autodidact and can't follow the part after "it is easily seen that"... which is the 31st equation: Shouldn't it be: $m_i\,{\bf r}_i\times \frac{d^2{\bf r}_i }{dt^2}= \frac{d}{dt}(m_i r_i ...
8
votes
1answer
175 views

Time derivative of the state vector as expressed in abstract Hilbert space vs. as a wavefunction

The Schrodinger equation in Hilbert space is expressed as : $$\frac{\partial}{\partial t} \psi(t) = \frac{-i}{\hbar}H\psi(t). $$ Here $\frac{\partial}{\partial t} \psi(t) \equiv \psi'(t) \equiv\lim ...
2
votes
1answer
53 views

Taylor series expansion of $\ln$ and $\cosh$ in distance fallen in time $t$ equation

I want to find the Taylor expansion of $y=\frac {V_t^2}{g} \ln(\cosh(\frac{gt}{V_t}))$ I have tried using the fact $\cosh x= \frac {e^x}{2}$ for large t, which works, I just need help on small values ...
0
votes
1answer
47 views

What is the infinitesimal work done when the force is given by the gradient of a scalar function that depends both on position AND time?

The title is slightly confusing but I didn't know how else to phrase my question. Basically, this is the situation: When the force applied to a particle is given by the gradient of a scalar function ...
3
votes
1answer
91 views

Contradiction of a scalar product

Can anyone resolve this contradiction: ...
1
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0answers
55 views

Index Notation Double Curl

My question is about Einstein notation. It does not matter the specifics of this example (the del operator could be another random vector), I just want to know if my assumption about notation is ...
3
votes
1answer
104 views

What is the difference between $\nabla _{\sigma} $ and $ \nabla^{\sigma}$?

What is the difference between: $\nabla _{\sigma} $ and $ \nabla^{\sigma}$? I've been told that the first is the covariant derivative, however I'm just starting a course on spacetime geometry and ...
2
votes
1answer
80 views

Are covariant derivatives of Killing vector fields symmetric?

I'm reading the Lecture Notes on General Relativity by Matthias Blau, and in section 9.1 (point 1) he writes: Let $K^\mu$ be a Killing vector field, and ${x^\mu(\tau)}$ be a geodesic. Then the ...
0
votes
1answer
89 views

Index Notation with Del Operators

I'm having trouble with some concepts of Index Notation. (Einstein notation) If I take the divergence of curl of a vector, $\nabla \cdot (\nabla \times \vec V)$ first I do the parenthesis: ...
7
votes
4answers
263 views

Conserved quantities and total derivatives?

I am having a bit of a crisis in understanding of the physical meanings of total derivatives. When a quantity $\rho$ (be it a vector or a scalar) is said to be conserved, then (mathematically) ...
1
vote
1answer
97 views

Question concerning the Feynman Lectures of Physics

I am reading the Feynman lectures and at this point http://www.feynmanlectures.caltech.edu/I_13.html#Ch13-S3 it says as follows: The time derivate of the potential energy is $\begin{equation} ...
2
votes
3answers
181 views

Ordering of differential operators

If we write something like: $\partial_a X_{\mu} \partial^a X^{\mu}$ Does that mean the first derivative is only applied to the first X? ($\partial_a X_{\mu})( \partial^a X^{\mu}$) Or is the ...
0
votes
1answer
69 views

How to derive the schwarzchild metric?

I'm having trouble differentiating the following when making a change of co-ordinates to determine the Schwarzchild metric. $$r'^{2}=r^{2}C(r)$$ Then taking the total derivative of both sides, the ...
0
votes
1answer
66 views

Fermion propagator is not a Grassmann-odd object?

Is the following differentiation correct: $$ \frac{\delta}{\delta\eta\left(z\right)}\int d^{4}yS_{F}\left(z-y\right)\eta\left(y\right) = S_F\left(z-z\right)$$ where $\eta$ is a Grassmann-valued ...
6
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4answers
394 views

Name this Mulltivariable Calculus Theorem

In Robert Wald's book General Relativity a multivariable calculus theorem is cited on page 16, which states: If $F:\mathbb{R}^n\mapsto \mathbb{R}$ is $C^{\infty}$ then for each $a=(a^1,...,a^n) \in ...
0
votes
1answer
64 views

Covariant derivative of a vanishing tensor component [closed]

Is the covariant derivative of a vanishing tensor component necessarily zero?
0
votes
1answer
27 views

Differentiate wave speed, don't understand

The speed $v$ of some wave is $ω/k$ and I want to differentiate this with respect to $k$. Apparently this equals: $dv/dk = d(ω/k)/dk-ω/k^2$ But I don't understand why. Isn't this just saying "the ...
1
vote
1answer
78 views

Covariant derivative as a tensor

$$\nabla_{j} v^{i}~=~g^{ik}\nabla_{j}v_{k}.$$ Does this equality involve an intermediate step, where I take the metric inside the derivative, and then use the fact that covariant derivative of the ...
0
votes
1answer
88 views

Help deriving the general linear wave equation $d^2y/dx^2=(1/v^2)d^2y/dt^2$ [closed]

How do I derive the General Linear Wave Equation $$d^2y/dx^2=(1/v^2)d^2y/dt^2?$$ My teacher differentiated the general wave function $f(x + vt)+g(x - vt)$ twice with respect to both variables to get ...
0
votes
2answers
63 views

Taking time derivative of two dependant variables

I'm not entirely sure if this is correct. I have to take the time derivative of the following: $$\frac{d}{dt}mr^{2}\dot{\phi}$$ Now, both $r$ and $\dot{\phi}$ depends on the time $t$, so I have to ...
0
votes
0answers
68 views

How to do this index notation differentiation?

I am studying classical Maxwell fields and I am stuck on this differentiating part. How can I derive the result given below ? $$\dfrac{\partial}{\partial(\partial A_{\mu}/\partial x_{\nu})} ...
6
votes
4answers
652 views

What is the current of a capacitor when the derivative of voltage is undefined?

This is from the textbook I am reading: I know this equation for capacitors: $$i=C\cdot \frac { dv }{ dt }$$ Here is my question: how can diagram (a) be allowed if the derivative of the voltage ...
0
votes
1answer
54 views

Finding the Lagrangian from the derivative of position

I have to find the Lagrangian for a system. In the point of interest I have come up with the following position coordinates: $$x = Rcos(\omega t)+\ell sin(\phi)$$ and $$y = Rsin(\omega t)-\ell ...
0
votes
0answers
66 views

Vector Derivative Transport Theorem Application

I have a position vector in frame A, the derivative of which I want to take relative to an observer in frame B. I apply the Vector Derivative Transport Theorem. The obtained velocity vector is left in ...
0
votes
1answer
112 views

Why do these equations result an incorrect unit for acceleration?

Hello everyone. Imagine an object moving around a certain point on a circular orbit. Magnitude of the velocity is constant during the motion ($|v|$). The orbit radius is $r$. (I'd better notice ...
2
votes
3answers
208 views

Physical motivation for differentiation under the integral

I am thinking about the mathematical process of "differentiating underneath the integral", i.e. applying the theorem $$\partial_s \int_{-\infty}^\infty f(x,s)\,dx=\int_{-\infty}^\infty \partial_s ...
10
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2answers
368 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 ...
2
votes
1answer
207 views

What is a covariant derivative in gauge theory?

I've been studying electroweak theory and you need to keep the Lagrangian covariant by introducing covariant derivatives. What is a covariant derivative? And what does it mean to keep the Lagrangian ...
3
votes
2answers
258 views

Derivative with respect to a vector is a gradient?

I've encountered in some books (and even completed an exercise from the Goldstein by using it), a strange notation that seems to work exactly like a gradient, I have tried to look for an explanation ...
4
votes
1answer
80 views

Higgs mechanism in QED

I'm trying to understand the Higgs mechanics. For that matter, I'm exploring the possibility of giving mass to the photon in a gauge-invariant way. So, if we introduce a complex scalar field: $$ ...
3
votes
1answer
152 views

Neglecting second order differentials

I am currently doing some Lorentz invariance exercises considering infinitesimal Lorentz transformations, and have been told to neglect second order differentials. It's not the first time I have come ...
8
votes
1answer
246 views

When motion begins, do objects go through an infinite number of position derivatives?

This might be a very vague and unclear question, but let me explain. When an object at rest moves, or moves from point $A$ to point $B$, we know the object must have had some velocity (1st derivative ...