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Physical significance of metric compatibility

There is a formulation of GR where metric compatibility follows from the equations of motion. If you think of the Einstein-Hilbert action as being a functional of both the connection and the metric (...
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Ricci Identity with Torsion Proof

Hint: You can, in a first step, expand the outer derivative (write $D_\nu Z^\sigma=A_\mu^\sigma$ if you wish). You will get a partial derivative acting on $A_\mu^\sigma$ and two terms with ...
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Derivation of entropy, I don't understand the relation $\frac{\partial S_2}{\partial E_1} = -\frac{\partial S_2}{\partial E_2}$

It's not unreasonable to be confused about this. Let's say you have a function $f$ of one variable and $f'$ is its derivative. Then we have $$\frac{d}{dx} f(c-x) = \color{red}{-}f'(c-x)$$ via the ...
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Ricci Identity with Torsion Proof

Remember that $\nabla_{\nu} Z^{\sigma}$ is a type $(1,1)$ tensor, so $\nabla_{[\mu} \nabla_{\nu]} Z^{\sigma}$ will spit out terms like $\Gamma_{[\mu \nu]}^{\lambda} \nabla_{\lambda} Z^{\sigma}$.
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Non-parallel light diffraction

Yes it does. A common version on the double slit experiment is, in fact, the one you describe. You can, for instance, install a narrow hole with the primary light source behind it. This hole will act ...
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What are the spaces in which quantum fields belong and how does that affect the hermitian conjugate of $\partial_{\mu}$?

First, let's consider the derivative $\partial$ as an operator on the Hilbert space $\mathcal{H} = L^2(\mathbb{R}^n)$ of square-integrable functions on $\mathbb{R}^n$ (the space typically encountered ...
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Ricci Identity with Torsion Proof

We have $$\nabla_\mu \nabla_\nu Z^\sigma=\partial_\mu(\nabla_\nu Z^\sigma)+\Gamma_{\lambda\mu}^\sigma \nabla_\nu Z^\lambda - \Gamma_{\nu\mu}^\rho \nabla_\rho Z^\sigma$$ as $\nabla_\mu$ is acting on ...
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Rigorous treatment for continuous mass systems

In discrete case we sum over particles, in continuous case we integrate over some space, physical or coordinate. When calculating e.g. coordinates of center of mass of some continuous body, summation ...
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Notation and Terminology Questions from Schwartz' QFT Book

Yes. OP is right. The LHS of eq. (3.30) is supposed to be a partial derivative. Yes. OP is right. The RHS of eq. (3.31) is supposed to be a total derivative. See also this related Phys.SE post.
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Velocities - Equation 1.46 of Goldstein 3rd edition

He is using the standard chain rule for partial differentiation from calculus. The partial derivative and the total derivative are not the same. See any calculus text such as one by Kaplan, Thomas, or ...
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