All Questions
Tagged with calculus differentiation
318 questions
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Variation of Torsion-Free Spin Connection
In the book 'Supergravity' by Freedman and van Proeyen, in exercise (7.27) it is written
To calculate [the variation $\delta\omega_{\mu ab}$ of the torsion-free spin connection], consider the ...
- 372
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2
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133
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Lagrangian total time derivative - continues second-order differential
In the lagrangian, adding total time derivative doesn't change equation of motion.
$$L' = L + \frac{d}{dt}f(q,t).$$
After playing with it, I realize that this is only true if the $f(q,t)$ function has ...
- 535
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1
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Taylor expanding a function of an operator?
I am trying to understand the following description in my quantum mechanics textbook:
Let $F(\hat{A})$ be a function of an operator $\hat{A}$. If $\hat{A}$ is a linear operator, we can Taylor expand $...
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2
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201
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What is difference between an infinitesimal displacement $dx$ and a basis one-form given by the gradient of a coordinate function?
In general relativity, we introduce the line element as $$ds^2=g_{\mu \nu}dx^{\mu}dx^{\nu}\tag{1}$$ which is used to get the length of a path and $dx$ is an infinitesimal displacement But for a ...
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Given a Postion-time curve/function, how do I find the time spent per unit position?
I have recordings of the position time curve for a given 1D actuator.
I'm trying to find out the time spent per unit length.
To get this relationship, I tried to take an example of a linear function:
$...
- 57
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93
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Schwartz "QFT and the Standard Model", eq. 15.59, derivative trick, deriving with a dot product
$$\frac{\partial }{\partial s}M(s)= \frac{p^{\mu}}{2s}\frac{\partial }{\partial p^{\mu}}M(s)\tag{15.59}$$
$$\ s=p^{2}$$
How does the derivative with respect to $s$ turn into the expression on the ...
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Quantum derivatives: quantum calculus and their classical limit
Jackson derivatives and their $q,p$-version are defined to be
$$D_qf=\dfrac{f(qx)-f(x)}{(q-1)x}$$
$$D_{q,p}f=\dfrac{f(qx)-f(px)}{(q-p)x}$$
When trying to go $q\rightarrow 1$ and $q\rightarrow p$ I ...
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What does it mean to differentiate a scalar with respect to a vector?
I am reading the special relativity lecture notes that I got from a professor of mine. It says that the Lagrangian is
$$L = \frac{1}{2}m|\dot{\boldsymbol{x}}|^2 - V(\boldsymbol{x}) \tag{1}$$
The notes ...
- 1,254
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$x$-derivative of the wave function and its conjugate [closed]
I saw that in order to show that the normalisability of a wave function does not depend on time, there is a necessary step in the calculation that says that:
$$\left(\Psi^*\frac{\partial^2\Psi}{\...
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1
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Lagrangian for 2 inertial frames where only Speed is different by small amount
In Landau & Liftshitz’s book p.5, they go ahead and writes down lagrangians for 2 different inertial frames. They say that Lagrangian is a function of $v^2$.
So in one frame, we got $L(v^2)$.
In ...
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2
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2k
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How to deal with differentials? [duplicate]
I am currently working on this. More specifically my question is about Problem 2.5 b). In the solution they get from
$$
Nd\mu=-SdT+VdP
$$
to
$$
N\Big(\frac{\partial\mu}{\partial N}\Big)_{T,V}=V\Big(\...
- 131
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3
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233
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Problem with the constant magnitude of vectors if the change in the same vector is perpendicular to it [duplicate]
Note: I am merely a highschool student attempting to self-study Classical Mechanics, some of the assumptions I make are perhaps wrong, so please bear with me. Thank you.
This while can be condensed ...
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70
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Expressing infinitesimal physical quantities
In physics class, my teacher demonstrated that in polar coordinates, an infinitesimal area involving radial length dr and infinitesimal angle dθ is equal to rdr dθ, since the area is roughly a square ...
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"$\delta^2 S$" confusion regarding "second variations" in stability conditions
As far as I am aware, for some function of $n$ variables $f$, $\delta^2 f$ represents the third term in its Taylor expansion.
So, I've encoutered the following expression in my thermodynamics book:
...
- 135
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92
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Bianchi identity in EMT [closed]
$ ∇_a∇_b F_{ab} = 0 $ ($F_{ab}$ Faraday tensor in EMT.)
proof is given by
"To see this, assume a Minkowski spacetime for simplicity and adopt
Cartesian coordinates, so that the covariant ...
- 81
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3
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93
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What does it mean to differentiate a differential quantity with respect to another quantity it is not dependent upon? [closed]
Recently, I came across this equation while solving a few integrations:
d(xy) = xdy + ydx
When I searched for its proof, I found that we assume another differential quantity, say dp, and then ...
- 53
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4
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261
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Show that $d\mathbf{v}^2/dt = 2\mathbf{v}\cdot d\mathbf{v}/dt$ using geometry only
I have just begun reading Modern Classical Physics by Thorne and Blandford and I am trying to wrap my head around their "geometric viewpoint" on classical mechanics. The first exercise in ...
- 510
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Finding back a simple SDE from its solution
I'm trying to self-learn Kurt Jacob's Stochastic Processes for Physicists: Understanding Noisy Systems. I've followed Chapter 3, where I saw how to derive that the solution to the SDE
$$
dx=\left(c+\...
- 11
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3
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198
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What is the definition of velocity?
We know that displacement is change in an object's position (here position means 'position vector'). Then velocity will be change in position of the object with respect to time, simply displacement/...
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150
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Does the gradient of potential energy exist independent of coordinates?
Potential energy $U(\vec{r})$ of a conservative force field $\vec{F}$ is defined as a function whose variation between positions $\vec{r}_A$ and $\vec{r}_B$ is the opposite of the work done by the ...
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Chain rule when the intermediary variable might be equal to zero
I came across the following question in the kinematics section of my introductory physics textbook:
The velocity of a particle moving along x-axis is given as $v=x^2-5x+4$ (in $m/s$), where $x$ ...
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45
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Physical and Diagrammatic representation of $a$=undefined when $v$=0 according to $a$=$vdv$/$dx$
$a$=acceleration
$v$=velocity
$x$=position along x axis
$t$=time instant
My teacher derived the $a$=$v$$dv$/$dx$ formula as follows
Assume a particle at time $t$ is at $x$ position having $v$ velocity
...
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2
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414
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Why does tangential acceleration become 0 when the velocity is max? [closed]
I know that tangential acceleration equal to zero when the circular motion is uniform, but why is it equal to zero, when the velocity is max or min? Because there is no relation between the value of ...
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46
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Partial derivatives and the Joule Coefficient
The Joule Coefficient for a van der Waals gas can be shown to be
\begin{equation}
\left(\frac{\partial T}{\partial V}\right)_U=-\frac{a}{C_VV^2}
\end{equation}
where $U$ is the internal energy of the ...
- 2,456
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65
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"To order $n$ of" arguments
Often one finds in physics textbooks that arguments will be made "to order $n$". I am not sure on what the procedure or argument ought to be when we have some denominator dependence though. ...
- 1,271
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Approximation of Small Perturbation [closed]
From Morin's Classical Mechanics, on the chapter of Small Oscillations in Lagrangian Mechanics, he does this approximation on the last equality, I don't understand what happened there.
I get the first ...
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261
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Infinitesimal coordinate transformation and Lie derivative
I need to prove that under an infinitesimal coordinate transformation $x^{'\mu}=x^\mu-\xi^\mu(x)$, the variation of a vector $U^\mu(x)$ is $$\delta U^\mu(x)=U^{'\mu}(x)-U^\mu(x)=\mathcal{L}_\xi U^\mu$$...
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196
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In physical calculations, is the elimination of higher-order small quantities an approximation or a strict equality in mathematics?
Physics sometimes uses a technique called the method of differentials, which seems magical and not very systematic. This makes me unsure which variable I should take the differential of, and sometimes ...
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Are terms tangential acceleration and normal acceleration only used for instantaneous velocity?
Are terms tangential acceleration and normal acceleration only used
for instantaneous velocity?
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How to transform a partial derivative to a directional derivative with respect to some affine parameter?
Suppose an affine parameter $\lambda$ is defined along a null geodesic with $dx^\mu/d\lambda=k^\mu$. How could I write the partial derivative $\partial f/\partial x^\mu$ by using $df/d\lambda$? If $k^\...
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Inverse of the covariant derivative
Given the covariant derivative of some tensor, for the sake of this example a covariant vector:
$$\nabla_\mu A_\nu$$
Is there a well-defined inverse operation on the covariant derivative such that it ...
- 613
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85
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Deriving smoothing kernels
I'm watching a video on smoothed particle hydrodynamics it just blindly claims that these smoothing kernels are pretty good.
$$W(r-r_b,h)\equiv\dfrac{315}{64\pi h^9}\left(h^2-|r-r_b|^2\right)^3$$
$$\...
- 114
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1
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$\nabla$, $\cdot \nabla$, $\nabla \cdot$, $\nabla^2$ - What do they do? [closed]
I'm trying to teach myself Smoothed Particle Hydrodynamics. Unfortunately, my background is in electronics, so the Navier Stokes equations are somewhat alien to me, as is vector calculus. The video I'...
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70
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What do equations involving infinitesimals say?
I am reading this note on the Bernoulli equations with the following derivations:
I am struggling to find a calculus based meaning for the above equations involving the infinitesimal $\delta V$: I ...
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3
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How do I write the gradient in angular coordinates ($\theta_1$, $\theta_2$, $\theta_3$)?
I have to find $\tau$ by finding the gradient of $U(\theta_1, \theta_2, \theta_3)$, where my coordinates are $(\theta_1, \theta_2, \theta_3)$. I assume the gradient is not the simple Cartesian ...
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6
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856
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How to understand instantaneous velocity concept [duplicate]
When I started learning instantaneous velocity it didn't make sense to me. I don't understand in real life why we can't measure instantaneous velocity and therefore why we use this concept.
Or is this ...
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2
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67
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Instantanous and uniform velocity and acceleration [closed]
If the mathemical expression of instantanous velocity is $d/t$, what is the mathematical expression of uniform velocity.
If the mathematical expression of instantanous acceleration is $v/t$, what is ...
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1
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172
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How does instantaneous velocity cause displacement in just one point? [closed]
I have a question.
Falling object graph is curve shape right?
And instantaneous velocity is tangent line but how does this velocity make displacement in distance? Because suppose instantaneous ...
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93
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Does car move when instantaneous velocity is zero? [duplicate]
In 3blue1brown: derivative paradox.
supposed car moving with:
$S(t) = t^3$
And velocity is:
$V(t) = 3t^2$
He asked when t = 0 velocity is 0 m/s , does that car move at that time ?
And here his ...
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Why did my rearrangement with chain rule end up equating velocity to position?
We all know acceleration is the time-derivative of velocity which in turn is the time-derivative of position. Vice versa: position is the integration of velocity and velocity itself is the integration ...
- 401
1
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187
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Lienard-Wiechert Potential derivation in Wald's "Advanced Classical Electromagnetism" [closed]
I want to follow the Lienard-Wiechert potential derivation in Robert Wald's E-M book, page 179. I do not understand $dX(t_\text{ret})/dt$ on the right side. I assume the chain rule is applied and $x'^...
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1
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How do I keep track of what to differentiate in a Dirac Hamiltonian/Lagrangian?
Suppose we have the dirac Hamiltonian:
$$
H = \int d^3y\bar\psi(y)_b(-i\gamma^k\partial_k+m)_{bc}\psi(y)_c.
$$
My question is should I think the derivative operator $\partial_k$ is acting on the ...
- 1,853
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1
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72
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Sine and Cosine Functions [closed]
So long story short, We were given a windmill to experiment with and a sensor could sense the Voltage produced and graph it concerning time. We decided to make a sine wave out of the positive and ...
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50
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Why area under a curve equals the sum of values of function/quantities taken elementally? [duplicate]
Background:
I was taught basic formulas of differentiation and integration when I started learning physics. However, it wasn't taught through intuitions and concepts. It was like : " Hey these ...
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1
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34
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Derivatives of the lagrangian of generalized coordinates [closed]
I know that
$$U= \frac{1}{2} \sum_{j,k} A_{jk} q_j q_k \quad \quad T= \frac{1}{2} \sum_{j,k} m_{jk} \dot{q}_j \dot{q}_k $$
and the lagrangian is
$$ \frac{\partial U}{\partial q_k} - \frac{d}{dt} \...
- 11
1
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2
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138
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The treatment of infinitesimal quantities [duplicate]
Please be advised that my question is different from some of the existing threads like this one.
I have long been convinced that if we are to question the value of something which we ultimately are ...
- 862
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1
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57
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What does this vertical line notation mean?
Here is the definition of the Noether momentum in my script.
$$I = \left.\frac{\partial L}{\partial \dot{x}} \frac{d x}{d \alpha} \right|_{\alpha=0} = \frac{\partial L}{\partial \dot{x}} = m \dot{x} = ...
- 85
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0
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91
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Derivation of a partial derivative equation by Albert Einstein in Special Theory of Relativity
I was reading Albert Einstein's "On the Electrodynamics of Moving Bodies". In section "Kinematical Part", on $3 (Theory of the transformation of coordinates and times from a ...
- 21
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0
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167
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Partial derivatives in thermodynamics: general mathematical procedure [closed]
In the lecture notes (thermodynamics) the following mathematical identity is often used:
$$ \left(\frac{\partial A}{\partial X}\right)_Z = \left(\frac{\partial A}{\partial X}\right)_Y + \left(\frac{\...
- 85
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1
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69
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Use Index Notation properly when indices are already used in identifying which bases is the matrix metric calculated with respect to
I am wondering how to apply the usual linear algebra to the rather unfamiliar case of 'matrices' with indices in special relativity or even general relativity. In particular, consider$$f=\sqrt{-\det\...
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