# Tagged Questions

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### What is a covariant derivative in gauge theory?

I've been studying electroweak theory and you need to keep the Lagrangian covariant by introducing covariant derivatives. What is a covariant derivative? And what does it mean to keep the Lagrangian ...
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### Is acceleration $a = s/t^2$, or $a = 2s/t^2$, or something third?

I'm having trouble understanding some of the stuff regarding movement in my introductory physics class (I never thought I'd say that...) Acceleration is defined as $a = \frac{s}{t^2}.$ Distance can ...
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### Are there general circuits that differentiate/integrate empirically?

Is it possible to construct simple circuits, that given a time-varying input, produce an output that represents the derivative or integral of the input with respect to time?
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### What are $\partial_t$ and $\partial^\mu$?

I'm reading the Wikipedia page for the Dirac equation: $\rho=\phi^*\phi\,$ ...... $J = -\frac{i\hbar}{2m}(\phi^*\nabla\phi - \phi\nabla\phi^*)$ with the conservation of probability ...
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### Time evolution in quantum mechanics

We know that an operator A in quantum mechanics has time evolution given by Heisenberg equation: $$\frac{i}{\hbar}[H,A]+\frac{\partial A}{\partial t}=\frac{d A}{d t}$$ Can we derive from this ...
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### Why and when do we differentiate or integrate equations in physics? [closed]

I'm an engineering student and none of my professors ever explained why do we use derivations and/or integrations in physics. So I have this task, it goes like: The object is moving in a positive ...
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### Dimensional interpretation of inverse gradient length $\frac{d}{dx} \ln(Y)$

Preliminary definition: inverse gradient length Let me first explain what I mean by that term. The inverse gradient length of some quantity $Y$ (often thermodynamic temperature $T$) $L_Y^{-1}$ is ...
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### Gradient in the Frenet-Serret coordinate

I was simply thinking that the gradient in the Frenet-Serret coordinate at a particular point is similar to the gradient in the Cartesian coordinate. I simply assumed that Frenet space is an ...
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### Is this equation $\nabla_a\sqrt{-g}=0$ correct? [duplicate]

Is the equation $$\nabla_a\sqrt{-g}=0$$ correct? Here $\nabla_a$ is the Levi-Civita connection, and $g$ is the determinant of metric $g_{ab}$. Apparently, we have $\nabla_ag_{bc}=0$, but I am not sure ...
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### Why do we do partial and not covariant differentiation with $x^{\nu}$?

Why when taking the velocity vector we make $$u^{\nu}=\frac{d}{d\tau}x^{\nu}$$ and not $$u^{\nu}=\frac{\nabla}{d\tau}x^{\nu}$$ where in the last equation I meant the covariant derivative. Why?
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### Neglecting second order differentials

I am currently doing some Lorentz invariance exercises considering infinitesimal Lorentz transformations, and have been told to neglect second order differentials. It's not the first time I have come ...
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### Why do we assume electromagnetic fields to be doubly differentiable? [duplicate]

It seems like the identities of curl of gradient, divergence of curl, and the simple derivations of electromagnetic waves from Maxwell equations all rely on the symmetry (interchangeability of their ...
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### Partial derivatives vs total derivatives in thermodynamics

The specific heat of a system is defined as $$C_z = T \left( \frac{\partial S}{\partial T} \right)_{z=\text{const}}$$ Sometimes however, I find the same definition, but with total derivatives ...
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### Why can't impulse be instantaneous?

We know from 2nd law of motion that $$\vec{F} = \frac{d\vec{p}}{dt}.$$ Now, a rate of change can be instantaneous. So, rate of change of momentum is instantaneous. But I doubt how can there be ...
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### Differentiation and delta function

Need help doing this simple differentiation. Consider 4 d Euclidean(or Minkowskian) spacetime. $$\partial_{\mu}\frac{(a-x)_\mu}{(a-x)^4}= ?$$ where $a_\mu$ is a constant ...
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### Differentiation of a vector with respect to a vector

Does differentiation of a vector with respect to a vector make any sense? Even if it makes sense, how does it make any physical meaning? I mean what is the physical interpretation?
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### Differential operators in curvilinear coordinates

In the appendix A of Griffith's Electrodynamics text, he cites Spivak's Calculus on Manifolds as a reference more a more complete treatment of taking the gradient, curl, divergence, and Laplacian in ...
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### Derivation of the Riemann tensor confusion

I'm trying to understand the derivation of the Riemann curvature tensor as given in Foster and Nightingale's A Short Course In General Relativity, p. 102. They start by giving the covariant derivative ...
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### Is this covariant derivative identity true?

Trying to work through a textbook derivation of the geodesic deviation equation, I've calculated this identity:$$u_{;\beta}^{\alpha}u_{\alpha}=u_{\alpha;\beta}u^{\alpha}.$$ If this is true, I'm ...
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### Meaning of $\nabla_{\mathbf{p}_k} W(\mathbf p, h)$ in PBF

I'm reading this paper on Position Based Fluids and I couldn't understand the meaning of $\nabla_{\mathbf{p}_k} W(\mathbf{ p_i - p_j}, h)$ in the equation 7 (see below). …the gradient of the ...
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### Do integrals of position make any sense? Do they have an application? [closed]

I know that taking the derivative of position with respect to time defines what we call velocity, but I've never heard of physicist going in the opposite direction with position. Is there any ...
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### Local Coordinate Expressions for Lie Derivatives

I'm currently working through the math chapters of Norbert Straumann's book on General Relativity. I have trouble understanding the coordinate expression of the Lie derivative of a basis vector. The ...
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### What does it mean to differentiate a spinor-valued field?

Peskin and Schroeder, equation 3.28, states that the Klein-Gordon equation $$(\partial^2+m^2)\psi=0 \tag{3.28}$$ is a valid choice of equation for a Dirac spinor field. Their explanation makes sense (...
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The D'Alembertian operator is defined as $$\Box = g^{\nu\mu}\nabla_\nu\nabla_\mu$$ For the Minkowski metric in Cartesian coordinates that is $$\Box=\frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \... 1answer 78 views ### Estimating divergence of set of vectors I have a set of points where directions and intensities of a flow are given (in 3D). Is it possible to estimate the divergence of the flow defined by those vectors? I only need a rough estimate and I ... 1answer 100 views ### Why do derivatives act on vector fields on a worldsheet? The covariant derivative of a vector A^{\mu} at a point x is defined as$$D_z A^{\mu}=\partial_zA^{\mu}+\Gamma^{\mu}_{\rho\sigma}(x)\partial_{z}x^{\rho}A^{\sigma} where Greek symbols are ...
Suppose $D\subset \Bbb R^3$ contains a fluid and that $f : D\times \mathbb{R}\to \mathbb{R}$ is a time dependent function defined on the fluid region. In that case, the material derivative is defined ...
I'm reading the Lecture Notes on General Relativity by Matthias Blau, and in section 9.1 (point 1) he writes: Let $K^\mu$ be a Killing vector field, and ${x^\mu(\tau)}$ be a geodesic. Then the ...