# 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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### 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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### Defining the covariant derivative on bitensors

Bitensors (tensors defined on two different points) are an extension of tensors found in some applications of general relativity, where objects such as the world function, parallel transport operator, ...
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### Partial derivatives vs total derivatives in thermodynamics

The specific heat of a system is defined as $$C_z = T \left( \frac{\partial S}{\partial T} \right)_{z=\text{const}}.\tag{1}$$ Sometimes however, I find the same definition, but with total derivatives ...
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### Is there a sense in which one can interpret $\delta'(0) = 0$?

I wasn't sure whether to post this under physics or math (and landed on physics due to fear of being crucified for lack of rigor on math.stackexchange). In field theory, when we encounter divergences ...
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### Lattice differentiation and Locality

Assume we define the locality of a theory in the following way: Assume we have a theory of real scalars, so this theory is non local if the action has terms like $$\int d^dx\,\phi(x)V(x-y)\phi(y).$$ ...
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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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### Derivatives in the Lorentz Transformation

I am trying to better understand the Lorentz Transformation on a fundamental level and gain some intuition of it. In the Lorentz Transformation, the derivative of x' with respect to x must be a ...
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### Explicit expression of gradient, laplacian, divergence and curl using covariant derivatives

For my course in General Relativity I am given the problem to find the expressions for the gradient, laplacian, divergence and curl in spherical coordinates using covariant derivatives. I have tried ...
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### Why does invariance commute with partial derivative?

This question applies more generally to actions, but I am going to ask it for a specific example. I am getting confused when considering the invariance of the superstring action under Weyl ...
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### Prove $D_\mu\phi^\dagger=(D_\mu\phi)^\dagger$

In gauge transformation, $D_\mu$ was defined to be $\partial_\mu-igA_\mu$. However, I have hard time to see that $D_\mu\phi^\dagger=(D_\mu\phi)^\dagger$ without ambiguity. (A comparable example in QED ...
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### 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 scale-...
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### Calculating the variation of an operator in two different ways

Let $$H_{T}=\dot{x}^{I}\frac{\partial}{\partial \psi^{I}}T(x,\psi)$$ and consider the transformation: $$x^{I}\mapsto x^{I}+i\epsilon\psi^{I} \\ \psi^{I}\mapsto\psi^{I}-2\epsilon\dot{x}^{I}$$ where ...
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### Lie derivatives and the tetrad formalism

I have been trying to learn about the tetrad formalism in general relativity and I understand the basic idea, but there is one issue that I can't seem to figure out: Is there a definition of a Lie ...
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### Looking for differential operators satisfying a specific commutation relation with the Laplace operator

Consider the Laplace operator on some manifold, $\Delta=(\det g)^{-1/2} \frac{\partial}{\partial x^j} \left( (\det g)^{1/2}g^{jk} \frac{\partial}{\partial x^k}\right)$. I am looking for differential ...
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### Is double derivative of a tensor component with respect to time itself a tensor?

Can you please clear my doubt that if we take double derivative of a tensor component with respect to time, will the resulting quantity be a tensor or not?
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### Invariance of differential operators

How do we prove that del operator is invariant under any kind of change of coordinates, specifically under galilean transformations? I am getting an extra term containing the relative velocity of two ...
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### What is the difference between zero and an infinitesimal number?

In a standard Atwood machine physics problem, the string going over the pulley is considered massless. So does that imply mass = 0 or mass = dm? General question: what is the difference between 0 and ...
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### Graph of $dQ/dt$ discontinuous

Charge is quantised then why how do we define $dQ/dt$ as current when graph of $Q$ v/s $x$ will be discontinuous and hence non-differentiable. Is it an approximation we use?
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I don't think context is needed but to make sure: I'm doing a homework exercise on binary system. P is the orbital period and E the energy of the system. The following is in the solution when trying ...
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### Lie derivative of the non-coordinate metric being 0

I'm trying to answer a question about a the Lie derivative of a metric in a non-coordinate basis. Here, the $C^{a}_{bc}$ are from the Lie derivative of one basis vector with respect to another, or ...
I am doing EMT and I am trying to calculate the divergence of this current density given as, $$\vec{J}(\vec{r}', t_r) = \vec{J}(\vec{r}', t - \frac{|\vec{r}-\vec{r}'|}{c})$$ for $\vec{r} = (x,y,z)$ ...