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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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How to pick a boundary layer coordinate or stretching transformation

I am following Introduction to Perturbation Methods by Holmes and am unsure how I to pick the power in my boundary layer coordinate if my governing equation is the Laplace equation given by \begin{...
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Time derivative of the continuity equation - accelerated flow

Given a fluid with accelerated motion $\vec a= d \vec v / d t$ (in one direction). The question is to write the continuity equation using the acceleration value. The continuity equation reads $$\frac{...
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Explanation of the identities $\rho=\rho(\delta)$ ($\delta$ function) and $\rho=q\,\delta(\bar{r})$

Poisson's equation is $$\boldsymbol{\nabla}^{2}\varphi=-4\pi k_{e}\,\rho, \tag{*}$$ that in the case of a point charge $q$, already with spherical symmetry, has as solution \begin{equation} \varphi(...
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Divergence of $ \frac{1}{s}\hat{s}$ in cylindrical coordinates

In Griffiths' electrodynamics, the divergence of $\frac{1}{r^{2}}\hat{r}$ is evaluated in spherical coordinates to be $4\pi\delta(r)$. I encounter the same problem in case of $\frac{1}{s}\hat{s}$ ...
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1answer
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Covariant derivative in field theory

I'm reading Physics from Symmetry by Jakob Schwichtenberg and in Chapter 7 he introduces the covariant derivative when deriving the interaction Lagrangian density for the spin-half - spin-1 field: $$ ...
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Galilean transformation and differentiation

Given $x=x’-vt$ and $t=t’$, why is $\frac{\partial t}{\partial x’}=0$ instead of $1/v$? $t$ seems to depend on $x’$ because if $t$ changes, $x’$ changes. Also, in this problem, $dx=dx’$ as well, but I ...
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Doubt in application of $GDT$ in electrostatics

Consider a volume charge distribution with continuous density $\rho({\bf r'})$. The electric field at ${\bf r}$ is: $${\bf E}({\bf r})=k\int_V \frac{\rho({\bf r'})}{R^2}\hat{\bf R}\, \mathrm dV$$ ...
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I could not proof that curl of gradient is zero. How can I do this by using indiciant notation? [migrated]

I could not find a way to equal this to zero. $\vec{\nabla}\times(\vec{\nabla}\phi)= \epsilon_{ijk}\partial_{j}(\partial\phi)_{k}= \epsilon_{ijk}(\phi(\partial_{j}\partial_{k}) + \partial_{k}(\...
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1answer
82 views

Can you apply product rule to arg of a bra-ket?

I found the following expression in a paper: $$ \frac{d}{dt}\arg\langle\phi_+|\dot{\phi_-}\rangle $$ where the $\arg$ term is the argument of the complex number given by inner product between two ...
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Cannot simplify expression with rotor and nabla with index notation [closed]

I need to handle simple operation which needs some skill in tensor algebra. I have to take $\mathrm{rot}$ from $ (\vec u \cdot\nabla)\vec u $. I am not very good at tensors operations, but I know ...
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Infinitesimal squared in metrics [migrated]

Metrics are often formulated by appealing to the square of an infinitesimal quantities. Examples of such are: $$ (ds)^2=(dx)^2+(dy)^2 $$ or $$ ds^2=dx^2+dy^2 $$ or $$ d(s^2)=d(x^2)+d(y^2) $$ ...
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Why does the negative sign arise in this thermodynamic relation?

I can't understand why $\left(\frac{\partial P}{\partial V} \right)_T=-\left(\frac{\partial P}{\partial T} \right)_V\left(\frac{\partial T}{\partial V}\right)_P$. Why does the negative sign arise? I ...
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Calculate heat production rates in well-mixed batch fermenter

I have a formula to calculate the heat production rates in well-mixed batch fermenter: $$V * c_p * \rho * \frac{dT_j}{dt} = F * c_p * \rho * (T_0 - T_j) + U * A * (T_R - T_j) + r_Q * V$$ U [kJ/($m^2$ ...
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1answer
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Notation for the divergence of a rank 2 tensor

I am studying advanced fluid mechanics and sometimes you see equations written in index notation like $$ Dv_i= \partial_t v_i +v_j\partial_jv_i$$ but sometimes you find this arrow/vector notation (...
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What is Lie-dragging? What is the relationship between Lie-dragging and Lie-derivative? [duplicate]

I found lie-dragging at following picture, i can not understand why L and N have property of (12)
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1answer
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What is the meaning of $d$? [duplicate]

What is the meaning of $d$? Is is Delta? If it is Delta, why is it then not $\Delta$? I am still confused with that. Can someone help explain it to me?
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Tensor Derivatives in Index Notation in Special Relativity

The energy-momentum tensor $T^{\mu\nu}$ is not uniquely defined because we can add a term $\partial_{\lambda}X^{\lambda\mu\nu}$ to it, where $X^{\lambda\mu\nu} = - X^{\mu\lambda\nu}$, and show that it ...
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Question about Lie derivative of connection [closed]

This is from Hughston and Tod, Ex 8.5. Please give me an idea how to start with this. If $\mathcal{L}_V g_{ij} = 2\phi g_{ij}$ then prove that $$ \mathcal{L}_V\Gamma^{i}_{jk} = \phi_{,j}\delta^{...
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Time derivative in rotating frame

In Goldstein (2ed) sec 4.9 - Rate of change of a vector, why does he say that the instantaneous angular velocity $\omega$ is not a derivative of any vector? $$ (d\textbf{G})_{space} = (d\textbf{G}...
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How to get the derivative of $F(a(x,y,t),b(x,y,t))$ wrt $t$ displayed as $f(x,y,t)(dF/dx) + g(x,y,t)(dF/dx)$? [migrated]

I have a function $F(a(x,y,t),b(x,y,t))$ whose derivative wrt t I would like to write as $$\frac{\partial F}{\partial t} = f(x,y,t)\frac{\partial F}{\partial x} + g(x,y,t)\frac{\partial F}{\partial y}...
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What is the justification for going from $\frac{d(v^2)}{dt}dt$ to $d(v^2)$ in this derivation regarding work in the presence of kinetic friction?

I do not understand what is being done in the last line of the quote, where we manipulate one integral to get another. Incorporating Newton's second law $\Sigma\mathbf{\vec{F}}=m\mathbf{\vec{a}}$ ...
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Reason why dot notation isn't used for time derivatives in Maxwell's equations [closed]

Maxwell's equations seem to be usually written: \begin{align} \nabla \cdot \mathbf{E} &= \rho/\epsilon_0,\\ \nabla \cdot \mathbf{B} &= 0,\\ \nabla \times \mathbf{E} &= -\frac{\partial \...
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1answer
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Lagrange equations in a conservative system, understanding $\nabla_i$

For a system of multiple particles with conservative forces: $\mathbf{F}_i = - \nabla_i V$, with $V \equiv V(\mathbf{r}_1,\dots,\mathbf{r}_N)$ the potential in function of the position of the $N$ ...
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Tangent vector to a curve [migrated]

I am trying to relate things simply. If a curve is on a flat 2D space represented by the parameter $\lambda$. In polar coordinate system $(r,\theta)$ at any lambda the tangent vector components are $$...
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1answer
34 views

Constraints and time derivative

Consider a system of $N$ particles. There are $C$ holonomic time independent constraints, $$ \begin{aligned} f_1(\mathbf{r}_1,\dots,\mathbf{r}_N) & =0 \\ f_2(\mathbf{r}_1,\dots,\mathbf{r}_N) & ...
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1answer
72 views

How to interpret $\left[f(A), B\right] = \left[A, B \right]\frac{\partial f}{\partial A}$ - how to differentiate with respect to an operator?

From here and here I know the commutation relation for two operators are: $$\left[f(A), B\right] = \left[A, B \right]\frac{\partial f}{\partial A}$$ if $\left[A, \left[A, B \right]\right] =0$ and $f$ ...
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366 views

General derivative of the exponential operator w.r.t. a parameter

I am interested in the calculation of the general $N$th derivative w.r.t. a parameter $\lambda$ of a quantum mechanical exponential operator with the following structure: \begin{equation*} \frac{\...
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1answer
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Question on Partial Differentiation in Thermodynamics

For energetic fundamental relation $U=U(S,X_1,\ldots)$ where $X_k$ represent extensive parameters $V$ or $N_j$, let \begin{equation}P_k=\frac{\partial U}{\partial X_k}.\end{equation} For entropic ...
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Does the proper four-acceleration $A^{\mu} = (0,0)?$

Let the proper four-position vector $x^{\mu}(\tau) = (0, \tau)$. Differentiating this successively wrt $\tau$ I get the four-velocity $u^{\mu}(\tau) = (0, 1)$ and then the four-acceleration $A^{\mu}(\...
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2answers
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How can we argue $\nabla\cdot v = 0$ when ${\mathscr r} \ne 0$ on this vector function?

I am dealing with the vector field: $$v = \dfrac{\hat{\mathscr r} }{{\mathscr r}^2}$$ And I am studying its divergence. If we compute it we get: $$\nabla\cdot\left(\dfrac{\hat{\mathscr r} }{{\...
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How can the following tensor can be reexpressed by totally symmetrized derivatives? See the attached picture

Particularly, in Wald paper https://doi.org/10.1063/1.528839 from tensor in (4) to (5).
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1answer
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Help with deriving simple heat equation [closed]

$$j^{q}=\frac{1}{2} n v[\varepsilon(T[x-v \tau])-\varepsilon(T[x+v \tau])]$$ To this: $$j^{q}=n v^{2} \tau \frac{d \varepsilon}{d T}\left(-\frac{d T}{d x}\right)$$ At first I was thinking of using ...
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1answer
58 views

Symmetry of the Batalin-Vilkovisky (BV) antibracket operation

Batalin and Vilkovisky define $^1$ an operation they call antibracket which is $$(F,H) = \Big(\frac{\partial_r F}{\partial \Phi^A}\Big) \Big(\frac{\partial_l H}{\partial \Phi^* _A} \Big) - \Big(\frac{...
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1answer
63 views

How does one deal with derivative operator in quantum field theory properly?

Given creation and annihilation operators, ${a^{\dagger}(x,t)}$ and $a(x,t)$ in non-relativistic quantum field theory, respectively, which satisfy the following properties: Now, I want to prove $$[...
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2answers
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Meaning of time derivative of an operator

Today when my professor was deriving this equation: $$\frac{\mathrm d\langle A\rangle}{\mathrm dt}=\frac{i}{\hbar}\langle\left[H,\,A\right]\rangle+\left\langle\frac{\partial A}{\partial t}\right\...
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1answer
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Hamiltonian time-independent, partial derivative always zero?

For conceptual simplicity, let's restrict the discussion to systems with a two-dimensional phase space $\mathcal P$ with generalized coordinates $(q,p)$. Hamiltonian is a function that maps a pair ...
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1answer
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Derivative of Lagrangian with respect to a vector

Sometimes to find an equation of motion, the Lagrangian is derivated with respect to the (position) vector. How can this be possible?
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2answers
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Gauge-covariance of the Yang-Mills field strength $F_{\mu\nu}^a$

Accordingly to Yang-Mills theories, after the introduction of a covariant derivative such that $$D_\mu = \partial_\mu - igA_\mu, \tag1$$ you can built the kinetic term for the gauge potential $A_\...
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Can we divide a vector by another vector? How about this: $a = vdv/dx?$

My physics teacher told us that we can’t divide vectors, that vector division has no physical meaning or significance. How about this: $$a = vdv/dx.$$ It says acceleration vector equals velocity (as ...
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Relation between computation of curl and divergence and their formal definitions

both divergence and curl of a vector field have a formal definition, however, we don't use these definitions when we compute the divergence or curl. so can we just derive the computations from the ...
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Instantaneous velocity applications

I refered these two questions Instantaneous velocity How to interpret instantaneous velocity using limit? and I understood how instantaneous velocity is defined. But why do we define it? Velocity ...
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3answers
115 views

Four velocity, acceleration, momentum and force in general relativity

I am getting slightly confused regarding the formal definitions of different four vectors in General Relativity. Many texts on relativity begin with four vectors and dynamics in Minkowski space, and ...
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1answer
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Confused about Navier-Stokes equation

Just look at the L.H.S of the compressible navier-stokes equation from wiki $$\rho(\partial_t \vec{u}+\vec{u}\cdot\nabla\vec{u})=...$$ How can I add a vector $\partial_t \vec{u}$ and a scalar $\vec{...
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2answers
87 views

Notation and concepts of Yang Mills Theory

I am studying loop quantum gravity using the book by Pullin and Gambini. I am having some trouble understanding and getting past the chapter on Yang Mills theory, mainly because I am confused about ...
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1answer
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Van der Waals cycle calculation [closed]

For the Van der Waals gas we get a cycle consisting of 2 isobaric and 2 isenthalpic processes. We are given $T_1$,$T_3$ and $v_1$,$v_3$. And we want to calculate the efficiency. Attempt of ...
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1answer
56 views

Differentiating Scalar along a geodesic

I have been studying GR for sometime and doing exercises from Schutz and I have a question about differentiating along a geodesic. Here is what I know. The equation of geodesic in terms of four ...
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1answer
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Confusion regarding the transformation law under diffeomorphism

While revisiting the transformation laws - under a diffeomorphism - of the partial derivative of a vector field I got confused by the following result. So, for a vector field $A$ the usual ...
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3answers
90 views

Divergence of magnetic field

Consider a point near one of the poles of a bar magnet. The magnetic field lines do appear to spread, but according to Maxwell's equations the divergence of a magnetic field is always zero. So what's ...
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2answers
61 views

4-Gradient vector and the Field strength tensor

Need some help evaluating the following 4-gradient, that of the gradient of the field strength tensor $$F^{\mu\nu}= \begin{bmatrix} 0 & -E_x & -E_y & -E_z\\\ E_x & 0 & -...