# 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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### 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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### Covariant derivative of the vielbein determinant

The vielbein postulate says that $$\nabla_\mu e_v^{\,a}=\partial_{\mu}e_\nu^{\,a}+\omega_{\mu\,\, b}^{\,\,a}\,e^b_\nu-\Gamma^\sigma_{\mu\nu}\,e^{\,a}_\sigma=0.$$ $\nabla$ is the coordinate covariant ...
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### Curl operator in Schwarzschild metric

I'm trying to write down the curl operator explicitly for a Schwarzschild metric in cylindrical coordinates. I am trying to use the general expression of the curl operator in orthogonal curvilinear ...
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### How is Infinitesimal coordinate transformation related to Lie derivatives?

I am reading the book "Gravitaion and Cosmology" by S. Weinberg. In section 10.9, while discussing Lie derivatives of tensors of different ranks, he makes a general comment: The effect of an ...
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### Doubt on $SU(2)_{L} \times U(1)_{Y}$ covariant derivative and its action on a fermion

I) Introduction I.1) The mathematical structure is quite clear: given a spacetime $M$, and a Lie group $G$ (the gauge group), we can construct the Principal bundle $P^{G}_{M}$. The connection $1$-form ...
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### What is the meaning of $\nabla _{\mu}\nabla _{\nu}\phi(r)$ in general relativity?

I know the covariant derivative of a tensor is $$\nabla_{\mu} V_{\nu}=\partial_\mu V_\nu-\Gamma_{\mu\nu}^{\lambda}V_{\lambda}$$ Now I want to obtain $\nabla_{\mu}\nabla_{\nu}\Phi(x)$ where $\Phi(x)$...
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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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### Bianchi identity of gauge theory

How to prove Bianchi identity? \begin{align*} \varepsilon^{\mu\nu\rho\sigma}D_{\nu}F_{\rho\sigma}=0 \end{align*} using Jacobi identity; \begin{align*} \epsilon^{\mu\nu\rho\sigma}[D_{\mu},[D_{\rho},D_{\...
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### Box size $R$ as a thermodynamic variable vs position radius $r$ inside the box

Titles are difficult but I hope I can do a better job in the text. I am working with a spherical box where I want to have the box's radius $R$ and temperature at the wall $T$ to be the main ...
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