# Tag Info

9

Below follows a handful of excerpts from the book Introduction to the Classical Theory of Particles and Fields (2007) by B. Kosyakov. Controversial/misleading/wrong statements are marked in $\color{Red}{\rm red}$. We agree with OP that the statements marked in $\color{Red}{\rm red}$ are opposite standard terminology/conventions. Some (not all) correct ...

6

I took a quick look at pages 59 and 60 of "Gravitation", section 2.6 "Gradients and Directional Derivatives", to see if there's anything there we can use to clarify this issue. In this section, the gradient of $f$ is $\mathbf df$, the directional derivative along the vector $\mathbf v$ is $\partial_{\mathbf v}f$ and the following relationship holds: ...

6

In the sense of "Copenhagen Interpretation", what exactly is an interpretation? What purpose does an interpretation serve? I would describe interpretations of quantum mechanics as part of the philosophy of physics. Here is a well-known quote by Bertrand Russell: "As soon as definite knowledge concerning any subject becomes possible, this subject ceases ...

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In normal usage, a gauge is a particular choice, or specification, of vector and scalar potentials $\mathbf A$ and $\phi$ which will generate a given set of physical force fields $\mathbf E$ and $\mathbf B$. More specifically, a physical situation is specified by the electric and magnetic fields, $\mathbf E$ and $\mathbf B$. A set of potentials $\mathbf A$ ...

5

Usually there's a great deal of overlap between the definitions of the momenta you've listed, so your confusion is understandable, but nonetheless there are cases (that I know of at least) where the distinction is more clearly enunciated: Momentum as known in Newtonian mechanics: The momentum is a vector quantity (its vectorial superposition for many ...

5

The first bullet would be read "$A$ dot $B$" or "The dot product of $A$ and $B$" The second bullet would be read "$A$ cross $B$" or "The cross product of $A$ and $B$"

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I believe this is just imprecise use of language by the author - there is nothing mysterious happening, it is just not well stated: As stated in the question, for a hypersurface $\Sigma$ defined by $$F(x) = c \in \mathbb{R}$$ we find that $$\mathrm{d}F = 0$$ must hold on $\Sigma$. This is crucial - it means that the 1-form $\mathrm{d}F$ acting upon ...

4

Personally, I think the primary reason for thinking about interpretations is that they can lead to new predictions. They generally don't do this directly - almost by definition, an interpretation makes no new predictions by itself - but by changing the landscape within which new theories can be proposed. The best example I know of is the Lorentz equations ...

3

An ADS search for "star formation" turns up about 142,000 articles with "star formation" in the title or abstract. The first article is a 43 page review paper of Star Formation in Galaxies in the Hubble Sequence, written by Robert Kennicutt, Jr, one of the leaders of the field. He never defines anything else to mean star formation and one of the "key words" ...

3

I will show this with a simple $Na^+$ and $Cl^-$ atom configuration. $Na$ atom loses an $e^-$ to become +vely charged. And the $Cl$ atom gains an electron to become -vely charged. Hence the charge on an electron is transferred.

2

Continuous symmetries of the action of a system which are global, that is, do not depend on where they act, give rise through Noether's theorem to conserved quantities. For example, a translation in time $t \to t+\epsilon$ for $\epsilon \in \mathbb{R}$ is a global transformation, and leads to energy conservation. On the other hand, if an action is invariant ...

2

The multi-dimensional analog of simple harmonic motion is an object subject only to a harmonic potential, $U = \frac 1 2 k ||\vec r - \vec r_0||^2$, where $k$ is a positive constant (oftentimes called the spring constant), $\vec r$ is the object's position, and $\vec r_0$ is the position of the center of the potential. By choosing the origin to be the center ...

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I believe it's the integral of intensity over time. $$\text{Lifetime intensity}=\int_0^\infty{I(t)dt}$$ "Intensity" in the instantaneous light output - the total light you get is that intensity from excitation until final decay. That said - it seems that the figure 2 in the paper you referenced in your link is using inconsistent labeling - because in the ...

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I believe you are confused because you are mixing up related but slightly different quantities. Yes, a partial derivative is a vector and yes, a vector is an object with an upper index. The above statement may seem contradictory, but in fact it is not for the following reason. A vector is an abstract quantity that is an element of a "vector space". In ...

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(I am dropping the bothersome factors of $\mathrm{i}$ in this answer, they contribute nothing to understanding what is going on) The gauge covariant derivative exists for all forms on the spacetime manifold $\mathcal{M}$ taking value in a representation of the gauge group. (Formally, these are sections of associated vector bundles to the gauge principal ...

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Actually, they are one and the same thing. Before I delve into the question you asked, let me quickly describe a closely related analogy - the covariant derivative in GR. This is a quantity $\nabla_\mu$ that acts differently on different objects. In particular  \nabla_\mu \phi = \partial_\mu \phi,~~~ (\nabla_\mu V)_\nu = \partial_\mu V_\nu - ...

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