# 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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### Laplace transform: How to evaluate partial derivative in the denominator of a fraction?

I am solving a differential equation using the Laplace transform. However, to evaluate it I need to evaluate some strange terms. Specifically, I have a partial derivative in the denominator of the ...
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### Why does $\delta \vec{r} = \delta \vec{ \theta} \times \vec{r}$?

Hello fellow physicists, I was trying to understand some behavior on rotating objects, specifically about the formula $\vec{v} = \vec{\omega} \times \vec{r}$. The Book (Marion, J. B. (1965). Classical ...
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### "Why is $n$ held constant when taking the time derivative in the course of the Van Kampen's system size expansion?"

I follow the notation used in "stochastic processes in physics and chemistry"(p.245) by Van Kampen. Left hand side of master equation is $$\frac{\partial P(n,t)}{\partial t}=\cdots.$$ We ...
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### Show that $dE/dt = -bv^2$ (Help with differentiation) [closed]

The question is: Show that $$dE/dt = -b (dx/dt)^2.$$ And the solution is: ...
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### Reduce multiplicative noise to additive noise with singular matrices

I have a stochastic differential equation as $$\dot{\textbf{X}}=\textbf{A}\textbf{X}+\alpha(t)\textbf{B}\textbf{X}-\alpha^*(t)\textbf{B}^T\textbf{X}$$ where $T$ ...
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### How do we get Momentum operator from linear momentum? [duplicate]

How do we get the momentum operator $p=-i(h/2π) ∇)$?
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### Finding the vector potential

$$\nabla\times\mathbf{B}=\nabla\times\left(\nabla\times\mathbf{A}\right)=\nabla\left(\nabla\cdot\mathbf{A}\right)-\nabla^2\mathbf{A}=\mu_0\mathbf{J}\tag{5.62}$$ Whenever I try to work this out and ...
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### Derivative of displacement in deriving expression for intensity of sound waves

I am currently working on deriving the expression for intensity of a sound wave: https://openstax.org/books/university-physics-volume-1/pages/17-3-sound-intensity The previously mentioned book states: ...
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### Cylindrical incompressible Navier-Stokes, are the derivatives commuative?

I am currently trying to understand a paper by Eckhardt et al. (https://doi.org/10.1017/S0022112007005629). In it, a transformation is performed on the cylindrical incompressible Navier-Stokes ...
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### Physical Interpretation of Flowrate Time Derivative of Incompressible Fluid in Variable Volume Vessel with Single Inlet

I am an academic researcher who studies fluid mechanics of the left ventricle (the primary chamber of the heart that actually pumps blood to the rest of your body). The majority of my work focuses on ...
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### Confirming an action is invariant under a supersymmetric transformation

I am studying chapter 9 of the book Mirror Symmetry, available here. My question is relating to page 156/157 where Supersymmetry is being introduced for the first time in QFT in 0-dimensions. We are ...
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### Del operator confusion [closed]

The very first thing my textbook says is that the Del operator is defined as: $$\vec{\nabla}=\vec{a}^i\nabla_i$$ Where $\nabla_i$ is the covariant derivative and " $\vec{a}^i$ is the curvilinear ...
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### What is the intuition or the derivation of covariant derivative?

I asked this question in mathematics but the answer I got was a bit too abstract for me so I hope that my fellow physicists can give me more of an intuition or an easier explaination of my question. ...
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### Looking for the geometric meaning of the curl of Killing vector fields

From Killing equation $$\nabla_\nu \xi_\mu + \nabla_\mu \xi_\nu = 0$$ it can be shown that $\nabla_\nu \xi_\mu$ is antisymmetric. From it we can construct an antisymmetric tensor $\mathcal{A}_{\mu\nu}$...
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### What is the gauge covariant derivative?

What is the gauge covariant derivative in layman's? Does it describe the kinetic energy of the gauge fields?
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### Integral of derivative of delta function gives strange answer [closed]

So I've been doing some QM and I keep coming across the following type of integrals: $$\int f(x) \frac{\partial}{\partial x} \delta(x-x') dx.$$ I know that I should integrate by parts but then I ...
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### Covariant derivative of metric determinant with torsion

I have some troubles taking the covariant derivative of the metric determinant with torsion. Let's suppose that we take a metric such that $\nabla_\mu g_{\nu\rho}=0$. My reasoning is the following. ...
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### Is the rate of change of duration a valid quantity?

I was wondering that, if the duration of a recurring event varies as time goes on, what would the magnitude of this quantity be measured in? For instance, if the time for an oscillation of a weighted ...
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### Cross factor for dependent terms in a differential?

How do you derive a cross factor to decouple differentials into independent differentials? For example: $$d(PV)= PdV+VdP$$ $$PV=\int{PdV}+\int{VdP}$$ Obviously dP and dV are related. Do you simply ...
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### Equality of variables for small values of time, when the time derivative of the variables are equal to one another

It is given that $\frac{ds}{dt} = \frac{d\theta}{dt}$, i.e. the time derivative of s and $\theta$ are equal to each other. Does it follow that for small values of $t$, $\Delta s ≈ \Delta \theta$? My ...
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### How do extreme points work in Statistical Mechanics?

Suppose that I have an $S,V,N$ ensemble. Every variable is a function of the other variable: $U(S,V,N)$, $S(U,V,N)$, $V(S,U,N)$ and $N(S,U,V)$. The functions are everywhere differentiable. But there ...
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### Why does $\vec{a}=\vec{\omega}\times \vec{r}$ as well as the velocity does?

Today I came in class and in one of the problems the teacher used $\vec{a}=\vec{\omega}\times \vec{r}$ which made me very confused because I don't know where it comes from, it seems pulled out of thin ...
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### How does the covariant derivative satisfy the Leibniz rule?

In Carroll's "Spacetime and Geometry", he states on page 95 (section 3.2) that the covariant derivative, $\nabla$, is a map from $\left(k, l\right)$ tensor fields to $\left(k, l+1\right)$ ...
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Considering a partial derivative of a vector field $w^a$ in x-direction (also called here 1-direction) I can write it as $$\frac{ \partial w^a}{\partial x^1 } = \partial_1 w^a - \Gamma^a_{1c} w^c + \... • 9,075 0 votes 1 answer 57 views ### How to understand the derivatives in wave equation? I am looking at the derivation of the wave equation, but I am stuck on the math. Specifically, in the following: How do they get the equivalence between \frac{\partial}{\partial z} (\frac{dg}{du}) = ... • 139 2 votes 6 answers 206 views ### Lagrangian - How can we differentiate with respect to time if v not a function of time? In the Lagrangian itself, we know that v and q don't depend on t (i.e - they are not functions of t - i.e., L(q,v,t) is a state function.) Imagine L = \frac{1}{2}mv^2 - mgq Euler-Lagrange ... • 515 2 votes 0 answers 49 views ### How does the divergence change under a change of frame (with geometric algebra)? I'm trying to prove equations (85) and (86) from Hestenes' paper Gauge Theory Gravity with Geometric Calculus (ResearchGate version).$$ \dot{\nabla}^\prime \cdot \dot{\underline{f}}(A) = J_f^{-1}[ (\...
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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 ...