# 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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### Derivation of covariant derivative

I'm currently doing Introductory QFT and was confused about the origin of the additional terms in the covariant derivate. My understanding is as follows: If we begin with the Dirac Lagrangian ...
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### Instantanous and uniform velocity and acceleration

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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### At what point does the deviation from the minimum of a function become of first order rather than second order? [migrated]

I was reading Feynman's page on the Principle of Least Action, and he stated the following: "That’s a possible way. But we can do it better than that. When we have a quantity which has a minimum—...
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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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### 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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### What is the correct formulation of momentum balance for a body of continuum?

What is the correct form of the momentum balance equation for a continuum body $\mathscr{B}$ whose particles are fixed, and occupies volume $V(t)$ at time $t$? \begin{align} &\frac{\mathrm{d}}{\...
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### Why differentiation of Fourier operator is difficult? [closed]

I have a question when I read some papers about physics-informed neural networks. In the paper of physics-informed neural operator, they said "it is non-trivial to compute the derivatives for ...
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### How to prove $\mathrm{Tr}[(\partial_\mu U)U^\dagger]=0$?

I am studying ChPT by referring to "A Primer for Chiral Perturbation Theory" by Stefan Scherer. I'm having a problem with the consideration of terms that appear in the Lagrangian. The ...
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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 ...
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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 ...
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### Proof of differentiate form of dynamical semigroups

I am studying some basics of the pure mathematical background for open quantum systems from Angel Rivas`s book which is "Open quantum systems, an introduction". Here is a theorem (Page 6, ...
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### How can a definite acceleration integral be useful in mechanics and why is an indefinite integral not used?

We have an acceleration function and in order to find the displacement function, it would be logical to take an indefinite integral 2 times. Then we would get a function. Why is it proposed here to ...
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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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### Invariance over Galilean transformation

I want to prove that the Wave Equation is not invariant under Galilean Transformation. I'm having a little trouble with it but this is my attempt. 1. First of all, what does it mean by "not ...
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### How is the Material derivative related to Hamiltonian and Lagrangian?

The material derivative is: $$\frac{Df}{Dt} = \frac{\partial f}{\partial t} + \dot{\vec{x}} \cdot \nabla f$$ Where $f(\vec{x}, t)$ could be a scalar field or a vector field. If we look at the ...
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### Proving the relation $\frac 1 2 \left[\nabla^2,r \right] = \frac 1 r + \frac \partial {\partial r}$ (quantum mechanics exercise) [closed]

I'm trying to prove this relation in my quantum mechanics exercise book $$\frac 1 2 \left[\nabla^2,r \right] = \frac 1 r + \frac \partial {\partial r}.$$ Here's my attempt: Expand the Laplacian ...
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### How to define differentiation of a time-dependent vectors with respect to a specific reference frame in a coordinate-free manner?

It is usual in classical mechanics to introduce the derivative of a time-dependent vector with respect to a reference frame. This is accomplished through the use of a basis that is fixed with respect ...
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### Divergence not defined

I’m currently working on the practice problems in Introduction to Electrodynamics by Griffiths. I got confused by the solution to this problem. What does “ill-defined divergence” even mean? I ...
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### How to calculate the correlation function like $\langle \partial_i \phi(x) \partial_j\phi(0) \rangle$ by Gaussian path integral?

From the standard text book about quantum field theory, we know that if we consider $$\mathcal{L}=\frac{1}{2}(\partial_{\mu} \phi)^2-\frac{m^2}{2}\phi^2,$$ the partition function of this Gaussian ...
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