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2
votes
3answers
162 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
0answers
34 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
48 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
votes
4answers
375 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
48 views

Covariant derivative of a vanishing tensor component [closed]

Is the covariant derivative of a vanishing tensor component necessarily zero?
-3
votes
0answers
51 views

Why does $d$ mean? [migrated]

What do the $d$'s mean? I've seen them in other formulas as well.
0
votes
1answer
23 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
55 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
67 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
56 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
55 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
432 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
47 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
47 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
96 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 ...
1
vote
3answers
154 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
votes
2answers
333 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
165 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
166 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
68 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
130 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 ...
7
votes
1answer
229 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 ...
1
vote
2answers
2k views

Derive vector gradient in spherical coordinates from first principles

Trying to understand where the $\frac{1}{r sin(\theta)}$ and $1/r$ bits come in the definition of gradient. I've derived the spherical unit vectors but now I don't understand how to transform ...
1
vote
0answers
103 views

Scale-invariant differential operator

For example, the differential operator Laplacian is $$\nabla^2 = \frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}.$$ My questions are: Is it scale-invariant? what is ...
0
votes
2answers
53 views

Feynman's subscript notation

Consider this vector calculus identity: $$ \mathbf{A} \times \left( \nabla \times \mathbf{B} \right) = \nabla_\mathbf{B} \left( \mathbf{A \cdot B} \right) - \left( \mathbf{A} \cdot \nabla \right) ...
1
vote
1answer
589 views

Gravitational force exerted by a rod on a point mass

I have doubts with the solution of a certain problem. I will give the entire solution below and will lay out my doubts as well. A point mass $m_1$ is separated by a distance $r$ from a long rod of ...
-2
votes
2answers
145 views

Why and how maximum force is $\frac{dF}{dx}=0$

In an certain question my teacher asked to find the maximum force. She said that the maximum force in electrostatics means $\frac{dF}{dx}=0$. Why is it like that?
0
votes
3answers
169 views

Meaning of “Gradient with respect to coordinates of particle” in SPH

I'm currently trying to implement a simple SPH simulation based on a variety of papers. However as I'm not a trained physicist nor mathematician I have a small issue with the following notation and ...
0
votes
0answers
102 views

What is difference between $\frac {dr}{dt}$ and $\frac {\partial r}{\partial t}$? [duplicate]

What is difference in physical meaning of partial time derivative and ordinary derivative of $r$? $$\frac {\partial r}{\partial t}\quad\text{and}\quad \frac {dr}{dt}.$$ I know that ordinary time ...
1
vote
4answers
722 views

When we take time derivative of a function of time, then is the result another function of time, again?

(I'll try to explain my question by one known example), for example where the velocity is a function of time v(t) then its time derivative (which is acceleration: $a=\frac {dv}{dt}$) is another ...
0
votes
1answer
87 views

In Newtonian pressure, what type of function is force?

This is pressure in Newtonian mechanics: $$P=\frac {dF}{dA}.$$ What does this mean? (Doesn't it mean that force is a function of area?) What type of function is force?
-9
votes
3answers
157 views

Is there any other mathematical tool to measure velocity, instead useing derivative? [closed]

To measure velocity we use derivative $$v=\frac {dr}{dt}.$$ Is the any other mathematical tool to do this?
0
votes
3answers
227 views

Which quantity gives the resistance of a component?

In a current vs potential difference graph, we can obtain the value of the resistance of the component. There are books that say gradient-inverse is the resistance and also books that say the value of ...
0
votes
1answer
116 views

Covariant derivative-Differential

I was trying to prove that the derivative-four vector are covariant. This can be proved only if you consider the time and space derivatives to be $\dfrac{\partial}{\partial ...
11
votes
2answers
2k 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 ...
1
vote
1answer
210 views

Arbitrary tensor covariant derivative

what are the rules for performing covariant derivatives on tensors of arbitrary rank? I found a few examples of Tensor derivatives: $$\nabla_{c} T^a {}_{b} = \partial_{c}T^a {}_{b}+ \Gamma^a{}_{cd} ...
0
votes
0answers
106 views

Why does the cross derivative of the partition function disappear here?

They state that the chemical potential in a canonical ensemble is given by: $$\mu = -kT \frac{\partial{\ln Z(N,V,T)}}{\partial{N}} \tag{1}$$ But if I use the definition of chemical partial (which I ...
0
votes
1answer
69 views

Is there any case where one would use, snap, crackle or pop? [duplicate]

As we all know, if you differentiate distance with reference to time, you get speed, and likewise, differentiating speed you get acceleration. However, if you keep differentiating, to the rate of ...
10
votes
1answer
425 views

Is there a “covariant derivative” for conformal transformation?

A primary field is defined by its behavior under a conformal transformation $x\rightarrow x'(x)$: $$\phi(x)\rightarrow\phi'(x')=\left|\frac{\partial x'}{\partial x}\right|^{-h}\phi(x)$$ It's fairly ...
1
vote
2answers
549 views

What is the common difference between partial time derivative and ordinary time derivative? [duplicate]

What is difference between partial and ordinary time derivative? for example: what is difference between $\frac {\partial v}{\partial t}$ and $\frac {dv}{dt}$? where the $v$ is velocity.
1
vote
2answers
132 views

What is path of light in the accelerating elevator?

Mathematically, (by mathematically I means by equations) what is path of light in the accelerating elevator? What is the difference between an ordinary derivative and covariant derivative (which is ...
1
vote
0answers
116 views

Implicit Differentiation, A doubt

$v=v_c(\tau, t)$ is a smooth function and suppose we have a relation $y_c(\tau,v_c;t)=0$ when $x_c$ is written in the form $x_c=c+ty_c(\tau,v_c;t)$, $c$ is real constant, $t$ is real number denotes ...
37
votes
3answers
2k views
3
votes
6answers
655 views

Is acceleration $a = s/t^2$, or $a = 2s/t^2$, or something third?

I'm having trouble understanding some of the stuff regarding movement in my introductory physics class (I never thought I'd say that...) Acceleration is defined as $ a = \frac{s}{t^2}.$ Distance can ...
1
vote
1answer
190 views

$\nabla({\bf u}^2)=2({\bf u}\cdot \nabla){\bf u} - 2(\nabla \times {\bf u}) \times {\bf u}$

Please see the next link: http://www3.kis.uni-freiburg.de/~peter/teach/hydro/hydro02.pdf In (2.13), he used: $$\nabla({\bf u}^2)=2({\bf u}\cdot \nabla){\bf u} - 2(\nabla \times {\bf u}) \times {\bf ...
5
votes
0answers
108 views

Is it correct to sum over either index of the metric the same way?

I don't know if the following is correct, i want to compute the following derivative $$\frac{\partial }{\partial (\partial_{\mu}A_{\nu})}\left(\partial^{\alpha}A^{\beta}\partial_{\alpha}A_{\beta} ...
4
votes
2answers
1k views

Derivatives of operators

How do derivatives of operators work? Do they act on the terms in the derivative or do they just get "added to the tail"? Is there a conceptual way to understand this? For example: say you had the ...
2
votes
1answer
123 views

Partial derivative potential energy of 'free' vibration

I have this rather mathematical question about the calculation of the partial derivative of a potential energy function given by: $$U(x_i)=\frac{1}{2}\sum_{i,j}\frac{\partial^2U(0)}{\partial ...
4
votes
1answer
196 views

Do partial derivatives commute on tensors?

For example; is $$\partial_{\rho}\partial_{\sigma}h_{\mu\nu} - \partial_{\sigma}\partial_{\rho}h_{\mu\nu}=0$$ correct?
2
votes
2answers
172 views

Are there general circuits that differentiate/integrate empirically?

Is it possible to construct simple circuits, that given a time-varying input, produce an output that represents the derivative or integral of the input with respect to time?