# 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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### Is my thinking correct for partial derivatives and tensors?

So I was transforming the affine connection and I ended up with a term like this: $$\frac{\partial^2 x'^a}{\partial x'^b \partial x^p}$$ where $x$ and $x'$ are two different coordinate systems ...
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### What does $\Delta$ stand for? [duplicate]

Newton’s first law states that $\Delta v=0$ unless acted on by an external force, $F_{\mathrm{net}}\neq0$. Can someone explain to me what the $\Delta v$ symbol means?
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### Meaning of time derivative of the Lorentz factor $\gamma$?

This question about the Lorentz factor $\gamma$ in special relativity. I know what $\gamma$ means and how to drive. I'm wondering if I have time derivative of $\gamma$, what dose it mean conceptually?
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### $\delta Q = dU + \delta W$. Why is it $dU$ while others are partial differentials? [duplicate]

It is the first law of thermodynamics for a very small change in the state of the system. It is in Heat thermodynamics and statistical physics by Brij Lal, Dr. N. Subrahmanyam, and P.S. Hemne.
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### Equation of distance and time

How is this equation derived? $$r = r_0 + ut + at²/2$$ where $r_0$ is the initial position of particle and $r$ is the position of the particle after all the motion it has undergone, $a$ and $t$ ...
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### Why the continuous arrangement of point masses (particles) at infinitesimal separations leads to a extended system?

I am basically talking in terms of Newtonian mechanics. The Newton's laws started with a good and easy assumption of particles as point masses. This assumption clearly reformed physics and a great ...
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### Torsion form and exterior covariant derivative

The torsion form can be defined as the exterior covariant derivative of a solder form, $\Theta=d_\omega\theta$. This derivative is always in the fundamental representation of the algebra $\mathfrak g$ ...
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### “Chain Rule” for functional derivatives in the context of a derivation of the geodesic equation by the stationary proper-time principle

I have been working on deriving the geodesic action via finding the stationary points of the proper-time integral for a massive point particle. Consider a space-time manifold $M$ ($\dim M=4)$ equipped ...
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### Momentum operator dot a vector

Why is $P \cdot A = A \cdot P -i\hbar\nabla \cdot A$? I was just replacing $P=-i\hbar\nabla$ so I didn't get the first term on the right side
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### Can I contract index in this expression?

I'm reading Carrol text on general relativity, on page 96 they arrive to the term \begin{equation} \frac{\partial x^{\mu}}{\partial x^{\mu '}}\frac{\partial x^{\lambda}}{\partial x^{\lambda '}}\frac{\...
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### Verifying the valdity of interchanging $d/dt$ and $\partial/\partial q^j$ in deriving the Lagrange Equation [duplicate]

Goldstein mentions that we can interchange the derivative operator with respect to time with the derivative operator with respect to qj. I am having trouble figuring out why this is possible. ...
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### What is the covariant derivative of a metric tensor $\nabla_{\mu} g^{\mu\nu}$ =?

What is the covariant derivative of a metric tensor, this particular one to be specific $\nabla_{\mu} g^{\mu\nu}$? Notice we've got repetitive indices here. Is it zero and has it got to do anything ...
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### Why is the partial derivative a contravariant 4-vector?

The contravariant partial derivative is defined as following: $$\partial ^\mu = \frac{\partial}{\partial x_\mu}$$ where the index $\mu$ runs from 0 to 3. A contravariant vector under Lorentz ...
Take two coordinates with $\mathbf r$ and $\mathbf r'$ and take a function $f(|\mathbf r - \mathbf{r'}|)$. In many electromagnetism derivations I see a conversion like this  \nabla_r f(|\mathbf r - \...