Questions tagged [differentiation]

Differentiation is the set of techniques and results from Differential Calculus, concerning the calculation of derivatives of functions or distributions.

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Particle paths - the distance moved by a particle in a velocity field

This question is is context to particle paths. Particle paths are trajectories of a given particle in the velocity field: $$\boldsymbol{u}(\boldsymbol{x},t)$$ A particle location at position $\...
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What can go wrong with applying chain rule to angular velocity of circular motion?

Lets say I have a circular motion, like this: I know that: $$\omega = \frac{\text{d} \phi}{\text{d}t}$$ Mathematically, what I am doing wrong, when I attempt to apply the chain rule in the following ...
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If $g_{ij}$ is a tensor of type $(0,2)$, what is kind of tensor is $\partial_{i}g_{jk}$?

Suppose $g_{ij}$ is a tensor of type $(0,2)$, then what type of object is $\partial_{i}g_{jk}$? Is it even a tensor, and if so, of what type? Is the $\partial_{i}$ still a differential with respect to ...
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How we can prove this vector identity?

I was trying to rederive the formula of the angular momentum of electromagnetic field, and all the steps are clear for me except this one which I took from "Photons and Atoms: Introduction to ...
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1answer
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Why is the first derivative of the time-dependent Schrödinger equation continuous? Where does it come from?

I was taught in first year physics that the first derivative of the time-dependent Schrödinger equation had to be continuous. However I was never taught (or at least I don't remember) the reason why. \...
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Tensors Differentiation

I know that $\frac{\partial x^{\mu}}{\partial x^{\nu}}=\delta^{\mu}_{\nu}$ but a few days back, I read somewhere that $\frac{\partial x_{\mu}}{\partial x^{\nu}}=\eta_{\mu\nu}$. Can someone help me ...
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How do I imagine why divergence of curl and curl of gradient is $0$? [migrated]

I tried watching several videos on YouTube, but I failed to gain intuition. I tried to solve it by myself by imagining water flow but I was unsuccessful and got stuck. How do I imagine why divergence ...
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2answers
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Derivative Operators on a manifold

I am having some trouble coming to terms with the notion of a derivative operator on a manifold. In Robert M. Wald's General Relativity, the definition in the textbook is given in terms of 5 ...
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Gradient, one-form and Sean Carroll

"A tensor (k,l) is a multilinear map from k dual vectors and l vectors to R (...) The gradient, ..., is an honest (0,1) tensor." These citations are retired from Sean Carrol Spacetime and ...
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Commutation relation of four vectors

I was trying to prove that: $$[P_\mu, J_{\rho \sigma}] = i(\eta_{\mu \sigma} P_\rho - \eta_{\mu \rho} P_\sigma) $$ $\textbf{Attempt}$ $$\begin{align} [P_\mu, J_{\rho \sigma}] = [P_\mu, x_\rho P_\sigma ...
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What's the difference between differentiation and derivation? [migrated]

The question is pretty much straightforward... I just don't get the difference between those two. Is there an easy way of understanding it?
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1answer
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Metric independent affine connections

Normally, the affine connections are objects that define parallel transport. In general Relativity they are the Christoffel symbols of the second kind. Consequently, they depend on the metric tensor. ...
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Best Calculus one book [duplicate]

I’m currently in my senior year of high-school. I’m planning to major in physics. I really enjoyed basic calculus but I really want to start studying it for real. I know university courses include ...
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Determining the partial derivative of a metric tensor

Im new to the Tensor Calculus and General Theory of Relativity, and I have one question. I want to determine the Christoffel symbols in FRW metric. This is the general equation of Christoffel symbols: ...
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2answers
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Showing that relative motion is independent of center of mass motion?

I am trying to understand the concept behind separating the center of mass motion and the relative motion in the Schrödinger equation for the Hydrogen atom. The idea is that the Hamiltonian given by (...
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2answers
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D'Alembertian of a delta-function of a space-time interval (i.e. on the light-cone)

How one differentiates a delta-function of a space-time interval? Namely, $$[\partial_t^2 - \partial_x^2 - \partial_y^2 - \partial_z^2] \, \delta(t^2-x^2-y^2-z^2) \, .$$ Somewhere I saw that the ...
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What is meant when we say that a differential takes on a certain value?

As far as i understand it, total differentials are linear maps that map vectors to numbers. In thermodynamics we encounter statements that a we have reached equilibrium when a total differential of a ...
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1answer
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Right derivative of Grassmann number and associated anti-commutation relation

I am reading chapter 3 of Anomalies in quantum field theory by Reinhold Bertlmann and I found one statement that I don't know how to prove. First of all he defined the right derivative on the ...
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Change of a scalar field/vector field [migrated]

In my book the following is written: The change of a scalar field $du$ in an arbitrary direction, given by an infinitesimal vector with an arbitrary direction $d\vec r$ is calculated: $$du=u(\vec r + ...
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Newton's Law of Cooling: $\delta Q$ or $\mathrm{d}Q$?

In this popular answer, I invoked Newton's Law of Cooling/Heating: $$\dot{q}=hA\Delta T\tag{1}$$ $$\dot{q}=\frac{\mathrm{d} Q}{\mathrm{d}t}\tag{2}$$ $$\dot{q}=\frac{\delta Q}{\mathrm{d}t}\tag{3}$$ $$\...
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Time Reversal Symetry

good day to you all! I have just been introduced to Time reversal symmetry and had a few questions regarding the topic with regards to differential equations. Let's say that you had the equation in ...
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2answers
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Total time derivatives and partial coordinate derivatives in classical mechanics

This may be more of a math question, but I am trying to prove that for a function $f(q,\dot{q},t)$ $$\frac{d}{dt}\frac{∂f}{∂\dot{q}}=\frac{∂}{∂\dot{q}}\frac{df}{dt}−\frac{∂f}{∂q}.\tag{1}$$ As part of ...
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Are there differences in notation for the d'Alembert operator?

On Wikipedia the d'Alembert operator is defined as $$\square = \partial ^\alpha \partial_\alpha = \frac{1}{c^2} \frac{\partial^2}{\partial t^2}-\nabla^2 $$ However, my professor uses the notation: $$ \...
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1answer
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Is the relation “slope=velocity” mathematically valid?

$\text{Slope= tan(angle with respect to positive X-axis)= scalar output}$ $\text{velocity= a vector }$ Source: Hugh D Young_ Roger A Freedman - University Physics with Modern Physics In SI Units (...
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Affine Connection

On page 74 of Weinberg's General Relativity textbook he writes the following: Equation 3.2.4: $$\Gamma_{\mu \nu}^{\lambda} \equiv \frac{\partial x^\lambda}{\partial \xi^\alpha}\frac{\partial^2 \xi^\...
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How can I derive the covariant derivative of the Ricci tensor using the Ricci scalar?

\begin{align*} R&=g^{\mu\nu}R_{\mu\nu}\\ \Rightarrow \nabla^{\gamma} R&=\nabla^{\gamma}(g^{\mu\nu}R_{\mu\nu})\\ &=\nabla^{\gamma}(g^{\mu\nu})R_{\mu\nu}+g^{\mu\nu}\nabla ^{\gamma}(R_{\mu\nu}...
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1answer
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What does $\mathbf{A}\cdot\nabla$ mean here?

What does $\mathbf{A}\cdot\nabla$ mean in an expression like $(\mathbf{A}\cdot\nabla)\mathbf B$? I found this in Griffiths’ Classical Electrodynamics book and cannot figure it out.
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2answers
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Covariant derivative of a one-form

I understood that the covariant derivative of a vector field is $$ \nabla_{i}v^j=\frac{\partial v^j}{\partial u^{i}}+\Gamma^j_{~ik}v^k $$ Then why is the covariant derivative of a covector field $$ \...
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1answer
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Is there a difference between instantaneous speed and the magnitude of instantaneous velocity?

Consider a particle that moves around the coordinate grid. After $t$ seconds, it has the position $$ S(t)=(\cos t, \sin t) \quad 0 \leq t \leq \pi/2 \, . $$ The particle traces a quarter arc of ...
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What does $\exp\left( ax\frac{d}{dx} \right)$ do on $\psi(x)$?

I'm trying to find out $$\exp\left(ax\frac{d}{dx}\right)\psi(x)= \ \ ? $$ I tried spending the exponential and then operating the derivatives one by one but I found no pattern. Besides, it gets ...
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1answer
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Visual representation of the curl of the vector potential!

I know that the electric field (a vector field) is the result of the gradient of the electric potential,which is a scalar field of the type : $\Phi$ : $\mathbb{R}^3 \rightarrow \mathbb{R}$. So the ...
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2answers
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Applying the exterior differential to the first law of thermodynamics [closed]

I'm working on an exercise for an advanced statistical physics course. The question I'm struggling with is this: $$TdS=dE+PdV-\mu dN\tag{1}$$ Write the first law $(1)$ as $dS=....$ Applying the ...
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Doubt of gauge covariant derivatives: how can I derive it?

In the context of general relativity (GR) it is necessary to introduce the notion of covariant derivatives. From the point of view of a basic introduction, we always start to deal with GR in a highly ...
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1answer
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Correlation function of time derivative of fields

Suppose you have calculated a two point function for a field $\phi$, and the result is some function of the positions (it can be a generic function, not necessary a function of the distance $x_1-x_2$):...
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1answer
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What is the meaning of word 'rate' in physics?

Often, I have seen in physics the rate of change of velocity or something like that in kinematics. And in question based on speed, time and distance. I would like to know the meaning of the word rate ...
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1answer
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Suppose for all value of $r$ expression for Effective Potential Energy $U_{eff}$ is zero, does that mean $F(r)$ is zero?

Suppose for all value of $\textbf{r}$ expression for Effective Potential Energy ($U_{eff}$) is zero, does that mean $F(\textbf{r})$ is zero?
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Rewriting an expression into a manifestly supersymmetric form

In $\mathcal{N}=2$ supersymmetry, one can define the following superderivatives \begin{equation} D_{\theta}=\partial_{\theta}+\frac{1}{2}\bar{\theta}\partial_{u}, \hspace{4mm} D_{\bar{\theta}}=\...
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Relations between partial derivatives used in the Maxwell relations [migrated]

In the context of Maxwell's relations, there are two frequently used expressions relating the partial derivatives of different thermodynamic quantities: $$\left(\frac{\partial x}{\partial y}\right)_{z}...
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1answer
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Gradient wrt. a Position Vector

I found the following equations in a dynamics textbook, The gravitational potential energy for any two particles in a $n$-particle system is given by, $$V_{ij} = - \frac {G m_i m_j}{r_{ij}}$$ where $...
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1answer
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Exterior Derivative on Curved Manifold (SpaceTime)

Considering the 2-Form Gauge Potential $B_{\mu \nu}$, we can write $dB=H$. In a flat manifold we have that $H_{\mu \nu \rho}=\partial_{\mu}B_{\nu \rho}+\partial_{\rho}B_{\mu \nu}+\partial_{\nu}B_{\rho ...
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2answers
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Is this notation inconsistent? If not, can some explain why not?

Im working through a textbook section on particle kinematics. An example given is relating vertical velocity to horizontal velocity and states: $y$ has a constant velocity of $10 \ \rm [m/s]$ $y=(0....
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What is the difference between the two time derivatives $\frac{\text{dL}}{\text{dt}}$ and $\frac{\partial \text{L}}{\partial \text{t}}$? [duplicate]

I am taking a course in classical mechanics, and I was wondering what the difference between the two time derivatives $\frac{\text{dL}}{\text{dt}}$ and $\frac{\partial \text{L}}{\partial \text{t}}$ in ...
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Can a numerical solution transport equation be stable AND have negative numbers?

I am trying do to a very simple fluid dynamics program using one dimensional transport equation: $$ \frac{\partial Q}{\partial t} = u \frac{\partial Q}{\partial x} - \alpha \frac{\partial^2 Q}{\...
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1answer
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Time derivative with respect to an observer moving with velocity $\mathbf{v}$

I am taking a class in fluid mechanics right now and my book has this statement with no explanation: What is the time derivate seen by an observer moving with a velocity $\mathbf{v}$ of a scalar ...
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1answer
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Lennard-Jones potential, distance $r$ for minimum energy

I'm sorry if the question seems stupid. I found (wikipedia) that the Lennard-Jones potential has it's minimum at a distance of $$r = 2^{\frac{1}{6}}\sigma.$$ If $U(r)_{min} = -\epsilon$ $$U(r) = 4\...
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2answers
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How do I properly partially differentiate with constant $p$ in thermodynamics?

I'm trying to solve the following problem: a one component system is described by the following equations $$U=\frac{A^2}{4}NT^2\exp \left(\frac{V^2}{N^2}\right),\qquad p=T^2f(v)$$ where $v = \frac{V}{...
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2answers
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Infinitesimal Changes - Notations

in my thermodynamics class we saw the following formulas: $$ dS = \frac{\delta Q}{T} $$ and $$ \delta W = PdV $$ This was part of a review of thermodynamics that we have seen previously; however, in ...
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1answer
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What is the analogy between the gauge covariant derivative and the covariant derivative in General Relativity?

A particle in the Dirac field can be described with the following equation $$i\gamma^\mu\partial_\mu\psi-m\psi=0$$ This is if the particle is non-interacting. However, if we impose a local $U(1)$ ...
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1answer
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Time derivative of $\rm{atan2}$ when $x=0$

I want to take the time derivative of the $\rm{atan2}$ function to calculate an azimuth rate in spherical coordinates, given position and velocity in Cartesian $xyz$ coordinates. $$\rm{atan2}(y, x) = \...
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1answer
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Covariant derivative of contracted tensor: why is it not obvious

In Wald's GR book (1984), he writes on page 221, In the timelike case, we restricted consideration to deviation vectors $\eta^a$ orthogonal to $\xi^a$ [$\xi^a$ is the normalized vector field of ...

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