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3
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1answer
447 views

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 ...
3
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6answers
4k views

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 ...
3
votes
2answers
233 views

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?
3
votes
2answers
326 views

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 ...
3
votes
1answer
734 views

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 ...
3
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1answer
211 views

Time derivative of time-translation Killing vector

I'm working with the spherically symmetric, static black hole metric. In the problem I'm working on, I'm told that $K$ is the time-translation Killing vector, $\frac{\partial}{\partial t}$ or $K = (1, ...
3
votes
1answer
778 views

Total and partial derivatives in thermodynamics and Maxwell relations

Consider the expression $$dS=\left(\frac{\partial S}{\partial T}\right)_VdT+\left(\frac{\partial S}{\partial V}\right)_TdV$$ I'm trying to understand how to derive an expression for $\left( \frac{\...
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2answers
4k views

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 ...
3
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1answer
163 views

Why do we need the material derivative?

I'm studying fluid mechanics, and I got the impression that the material derivative is nothing more than "differentiating along a path" and so I got confused on why do we need it. Basically, let $D\...
3
votes
2answers
46 views

Variation of Lagrangian density $\mathcal{L}$ w.r.t $x_\mu$

If a function $f(x(t),y(t))$ has no explicit dependence on the variable $t$, then $\frac{\partial f}{\partial t}=0$. In quantum field theory, the Lagrangian density $\mathcal{L}(\phi,\partial_\mu\phi)...
3
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1answer
28 views

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 ...
3
votes
1answer
75 views

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 ...
3
votes
1answer
126 views

Why is the gauge potential $A_{\mu}$ in the Lie algebra of the gauge group $G$?

If we have a general gauge group whose action is $$ \Phi(x) \rightarrow g(x)\Phi(x), $$ with $g\in G$. Then introducing the gauge covariant derivative $$ D_{\mu}\Phi(x) = (\partial_{\mu}+A_{\mu})\...
3
votes
2answers
54 views

How to derive wave speed/tension relation for the vibrating string?

I was studying vibrating strings and in my teacher's notes I found that, generically, if I change the tension on the string by $\Delta T$ then, the speed percentage change can be written as: $\frac{\...
3
votes
1answer
45 views

Derivative with respect to a difference of independent variables

I am dealing with an equation from nonlinear acoustics (Khokhlova-Zabolotskaya-Kuznetsov equation) where a strange term (for me as a mathematician) is used. The equation looks like this $$ \frac{\...
3
votes
1answer
55 views

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 ...
3
votes
1answer
135 views

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?
3
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1answer
294 views

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 ...
3
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0answers
40 views

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 ...
3
votes
1answer
76 views

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 ...
2
votes
4answers
1k views

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 ...
2
votes
4answers
1k views

Which Schrödinger equation is correct?

In the coordinate representation, in 1D, the wave function depends on space and time, $\Psi(x,t)$, accordingly the time dependent Schrödinger equation is $$H\Psi(x,t) = i\hbar\frac{\partial}{\...
2
votes
2answers
11k views

Derive vector gradient in spherical coordinates from first principles

Trying to understand where the $\frac{1}{r sin(\theta)}$ and $1/r$ bits come in the definition of gradient. I've derived the spherical unit vectors but now I don't understand how to transform ...
2
votes
1answer
2k views

How is the second-order covariant derivative of a scalar computed?

What is second-order covariant derivative $$\nabla_i\nabla_jf(r)$$ in terms of $r,\theta, g(r)$ and partial derivative, given that the metric takes the form $$ds^2=dr^2+g(r)d\theta^2$$ and $f$ is a ...
2
votes
1answer
335 views

How is it possible to pull out derivatives of a wavefunction?

In an early derivation, the following equation was stated: $$\frac\partial{\partial t}\lvert\psi\rvert^2 = \frac{i\hbar}{2m}\biggl(\psi^*\frac{\partial^2\psi}{\partial x^2} - \frac{\partial^2\psi^*}{\...
2
votes
3answers
321 views

How do I know what variable to use for the chain rule?

In my textbook the tangential acceleration is given like this: $$a_t=\frac{dv}{dt}=r\frac{dw}{dt}$$ $$a_t=rα$$ I understand that the chain rule is applied here like this: $$a_t=\frac{dv}{dt}=\frac{...
2
votes
4answers
575 views

Poincare invariant Lagrangians

The Lagrangian density of a Poincare invariant theory should not depend explicitly on the space-time coordinates. Does this mean $$ \partial_\mu \mathcal{L}=0~? $$ If this is the case doesn't the ...
2
votes
2answers
352 views

Any difference between thermodynamic double-derivative and derivative “at constant” value?

Reading about the Maxwell relations has left me confused, and I want a basic sanity check regarding the notation. The Wikipedia article breezes over the following switch of notation without really ...
2
votes
3answers
198 views

Ordering of differential operators

If we write something like: $\partial_a X_{\mu} \partial^a X^{\mu}$ Does that mean the first derivative is only applied to the first X? ($\partial_a X_{\mu})( \partial^a X^{\mu}$) Or is the ...
2
votes
1answer
56 views

What does the zero in the differential operator $\partial_0$ mean?

I have noticed the differential operator $\partial_0$ in a lot of equations whilst studying quantum field theory. I am used to the notation $\partial_x$ meaning $ \frac{d}{dx} \\\\ $ etc. but just a ...
2
votes
2answers
95 views

Lack of rigour in usual derivation of Work-Energy Theorem

The derivation of the Work-Energy theorem usually goes as follows: You define the work done on a particle under net force $\vec{F}$ as $$W=\int\limits_C \vec{F}\cdot\mathrm{d}\vec{r}$$ And then you ...
2
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2answers
141 views

Derivation of velocities in the Coriolis force

In Fitzpatrick's Newtonian Dynamics book on the Coriolis force, he states \begin{align} v_{x'}&\simeq V_0\cos\theta-2\Omega t V_0\sin\lambda~\sin\theta \tag{433}\\ v_{y'}&\simeq-V_0\sin\...
2
votes
3answers
80 views

Can we measure rates in real time?

I know what it means to say that my position is "X" at a particular moment in time. I can easily take a picture of my motion and observe my exact location at the instant the picture was taken. That is ...
2
votes
1answer
360 views

Partial derivative potential energy of 'free' vibration

I have this rather mathematical question about the calculation of the partial derivative of a potential energy function given by: $$U(x_i)=\frac{1}{2}\sum_{i,j}\frac{\partial^2U(0)}{\partial x_i\...
2
votes
1answer
419 views

Differentiation and delta function

Need help doing this simple differentiation. Consider 4 d Euclidean(or Minkowskian) spacetime. \begin{equation} \partial_{\mu}\frac{(a-x)_\mu}{(a-x)^4}= ? \end{equation} where $a_\mu$ is a constant ...
2
votes
1answer
212 views

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?
2
votes
2answers
197 views

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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votes
2answers
565 views

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 ...
2
votes
1answer
84 views

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 ...
2
votes
1answer
103 views

Can these two terms cancel out?

In trying to prove that $$\Gamma_{\mu\nu\lambda} = \eta_{ab}J^a_bJ^b_{\nu\lambda}.$$ The author canceled out while expanding the first equation $$J^a_{\mu\lambda}J^b_\nu$$ with $$J^b_{\mu\lambda}J^a_\...
2
votes
3answers
260 views

Physical motivation for differentiation under the integral

I am thinking about the mathematical process of "differentiating underneath the integral", i.e. applying the theorem $$\partial_s \int_{-\infty}^\infty f(x,s)\,dx=\int_{-\infty}^\infty \partial_s f(x,...
2
votes
1answer
43 views

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 ...
2
votes
2answers
86 views

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 ...
2
votes
1answer
50 views

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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2answers
83 views

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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1answer
91 views

How can the D'Alembertian of a field be interpreted intuitively?

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} - \...
2
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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 ...
2
votes
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 ...
2
votes
1answer
204 views

Relationship between Connection and Material Derivative

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 ...
2
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1answer
331 views

Are covariant derivatives of Killing vector fields symmetric?

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 ...