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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Question on notation convention regarding partial derivatives

H.Risken's book "The Fokker-Planck Equation" contains the following formula for the general 1D Fokker-Planck equation: $\frac{\partial W}{\partial t}=\left[-\frac{\partial}{\partial x}D^{(1)}(x)+\...
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Time taken by a boat to stop with its engine turned of moving in a lake [closed]

The question is taken from I.E.Irodov and it states that A motorboat of mass $m$ moves along a lake with a velocity $V_o$. At the moment $t=0$ the engine of the boat is shut down. Assume the ...
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Change in areal element

I am reading Griffith's Introduction to Electrodynamics., On example 1.7 while calculating surface integral of $x = 2$ for a cube of side 2., the book states $da = dy \cdot dz$ I don't get this, what ...
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Partial differentiation of action function [closed]

I'm trying to understand the principle of least action, and the author of the book I'm reading is presenting of derivation of the Euler-Lagrange equation. The author picks an arbitrary position $x_8$ ...
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3answers
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Verify that the electrostatic potential satisfies the Poisson equation [closed]

I'm reading Sect1.7 of Jackson's classical electrodynamics but I have trouble following his argument. Could someone help explain how exactly the Laplacian is evaluated in 1.30? Is it calculated with ...
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41 views

Commutator of covariant derivatives to get the curvature/field strength

For notation and convention, please see Gauge theory formalism and Generalizing the covariant derivate for gauge theory. The covariant derivative can be used to construct curvatures (called field ...
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33 views

Path Coordinates: direction problem (doubt) in derivative of tangential vector

Why is the direction of derivative of tangential vector perpendicular to the direction of the tangential vector?
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35 views

Dimensionless expression for differential equation

I am working through Nonlinear Dynamics and Chaos by Steven H Strogatz. In chapter 3.5 (overdampened beads on a rotating hoop), a differential equation is converted into a dimensionless form. I am ...
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72 views

Regarding directional derivatives [closed]

we know directional derivatives are the rate of change of any given scalar field along the given direction, and it is also equal to scalar product of gradient of the field and the unit vector along ...
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1answer
71 views

Continuity Equation for fluid in a curved spacetime

The current of fluid is the vector $J^{\nu}$. In free-falling laboratory due to Equivalence principle holds the known Continuity Equation $\partial_{\nu}\,J^{\nu}=0$, where the ordinary 4-divergence ...
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23 views

Converting density=mass/volume to relative rate equation for integration

In order to get the mass of an object from density, we might use \begin{equation} m = \int\rho(x)dx \tag{1} \end{equation} I understand why this works on a conceptual level, but I would like to be ...
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96 views

Equating $\partial_{t_r}$ = $\partial_{t}$ in the retarded potentials?

Im reading Griffiths E/M (4th edition) and came across something I don't understand: Page 446, footnote 8 which reads: "Note that $\frac{\partial }{\partial t_r} = \frac{\partial }{\partial t}$ since ...
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Generalizing the covariant derivate for gauge theory

Concrete example gauging the complex scalar field $\mathcal{L}=(\partial_\mu \phi)(\partial^\mu \phi^*)+m^2 \phi^*\phi$ $\phi(x) \rightarrow e^{-i\Lambda(x)}\phi(x)$ $A_\mu \rightarrow A_\mu + \...
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Guoy method derivation, or, how not to break math?

I start from $d\vec{F}=\nabla(\vec{m}\vec{B})$, $\vec{M}=\frac{d\vec{m_i}}{dV}$, and $\vec{M}=\frac{\chi_v}{\mu_o}\vec{B}$. I write out the definition of the force, determine we're only interested ...
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59 views

How can I take the Fourier Transform in the derivation of the Lorentz model? [closed]

How would one take the Fourier Transform of $$ m\frac{d^2x}{dt^2}+m\omega_0^2 x + m\Gamma \frac{dx}{dt} = -eE $$ to get $$ -m\omega^2 x+m\omega_0^2 x + jm\omega\Gamma x = -eE $$ This is in our class ...
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40 views

Evaluation of Hamiltonian of a charged particle under EM field

The Hamiltonian of a charged partical in EM field is given by $$H = \frac{\pi^2}{2m} -e \phi$$ where $$\boldsymbol{\pi}=-\mathrm{i} \hbar \boldsymbol{\nabla}+e \mathbf{A}.$$ To evaluate $\pi^2$, we ...
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Derivation of $x \partial_y - y\partial_x = \partial_{\phi}$

On a $S^2$-sphere we can define the coordinates $$x = \sin(\theta)\cos(\phi)\\ y = \sin(\theta)\sin(\phi)\\z=\cos(\theta).$$ Then I want to show that $$x \partial_y - y\partial_x = \partial_{\phi}.$$ ...
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Work done as change of potential, how total derivative is converted to partial derivative

I am reading Goldsetein's Classic Mechanics 3rd edition in Chapter 1 it says, If work done in moving form point 1 to 2 denoted by $W_{12}$, is independent of the path it should be possible to ...
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About Lagrange equation [duplicate]

$$\frac{\mathrm{d}}{\mathrm{d}t} \left ( \frac {\partial L}{\partial \dot{q}_j} \right ) = \frac {\partial L}{\partial q_j}.$$ I don't understand partial derivative by "function" (e.g. $q_j$). $q$ ...
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Meaning of the $\delta$ notation [duplicate]

Two examples I've seen of this so far are in statistical mechanics, when looking at the work done on a system: $$dW=-\frac{1}{\beta}\frac{\partial \ln{Z}}{\partial V}\delta V$$ and in the Noether ...
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Why is the divergence of the field zero in Maxwell's equations?

I read in a book called Vector Analysis by Murray R. Spiegel by Schaums Series, and I found that there is somewhere printed that the divergence of the electric field is zero. Since my teacher told ...
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Why the Lagrangian doesn't have an explicit time dependence?

I have a simple question regarding an example presented by Leonard Susskind and George Hrabovsky in their book on Classical Mechanics The Theoretical Minimum. In page 151, they state: "If there is ...
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366 views

What does it mean when we say 'The difference between two quantities is of first order'?

This question is about the explanation below Eq.(6.19) of Modern Quantum Mechanics by Sakurai Nepolitano (2nd edition) Let ${\bf j}(dx)$ be an operator that translates a point $x$ to $x+dx$. jf(x) = ...
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Why does $\vec{B}\cdot\frac{\partial \vec{B}}{\partial t}=\frac{1}{2}\frac{\partial}{\partial t} (B^2)$?

$$\vec{B}\cdot\frac{\partial \vec{B}}{\partial t}=\frac{1}{2}\frac{\partial}{\partial t} (B^2)$$ Griffiths states this result in his derivation of the Pontying vector, but I have absolutely no idea ...
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Does incomplete differentials $\delta Q$ or $\delta W$ have potentials? [closed]

I am very confused because my text book have following formula. $$dU = \delta Q \tag {1-1}$$ $$dU = \delta W \tag {1-1'}$$ Because these might mathematically mean "incomplete derivative = ...
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1answer
67 views

Prove $D_\mu\phi^\dagger=(D_\mu\phi)^\dagger$

In gauge transformation, $D_\mu$ was defined to be $\partial_\mu-igA_\mu$. However, I have hard time to see that $D_\mu\phi^\dagger=(D_\mu\phi)^\dagger$ without ambiguity. (A comparable example in QED ...
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51 views

What exactly is the Leibnitz rule in General Relativity?

In the Differential Geometry part of a course in General Relativity (for instance in David Tong's notes here in page 99, accessed 21 Nov, 2019), one frequently comes across the Leibnitz rule when ...
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35 views

Berry phase for effective gauge potential

On page 290 of Wens QFT he says that for the adiabatic motion of a single quasiparticle, for small t, $$ \left\langle\Psi^{h}\left(\xi(t+\Delta t), \xi^{*}(t+\Delta t)\right) | \Psi^{h}\left(\xi(t), \...
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What vector field property means “is the curl of another vector field?”

I'm an undergraduate mathematics educator and I teach a lot of multivariable calculus. I posed this question on MSE over four years ago and I haven't gotten any definitive answers (despite 12 upvotes ...
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Average velocity and instantaneous velocity

In some books of Physics in Italian language, they write that the instantaneous velocity $v$, is: $$v=\frac{dr}{dt}=\lim_{\Delta t \to 0} \frac{\Delta r}{\Delta t}$$ where $v_{\text{avg}}={\Delta r}/...
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Why does my answer vary?

Q)A wave moves with speed 300 m/s on a wire which is under the tension of 500N.Find how much tension must be changed to increase the speed to 312m/s. My method: Since $v= \sqrt{T/μ}.....(i)$,where T ...
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1answer
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A basic question about how to apply the gauge covariant derivative in Yang-Mills theory

I am sorry if this question is too stupid... We know that Yang-Mills equation (without source) can be written as $$D^\mu F_{\mu\nu}=0,\tag{1}$$ where $$D^{\mu}=\partial^\mu-ig A^{\mu}$$ and $$A^\mu=A^...
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Why does the derivative become $x=r\dot\theta\cos(\theta)$ when calculating angular velocity?

This is not a homework question because I do not want help with solving my homework. I would rather want an explanation of why the derivative of this seems to break math as I know it. Background ...
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Why do we differentiate displacement to get velocity if displacement is given as a function of time [closed]

Like if the displacement is given as S=(2t³)and if we are asked to find the velocity in 2 seconds then if we put t=2 in the expression we get 16 which isn't correct. The correct must be dS/dt=6t² and ...
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Related to the information contained in $a = v \frac {dv}{ds}$

While studying kinematics I came to the definition of acceleration which is $a = \frac {dv}{dt}$. But from this equation we can derive that $ a = v \frac {dv}{ds} $ which when I evaluate at $v=0ms^{-1}...
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From Maxwell equations to wave equation: what is the meaning of the differentiation?

Starting from the Maxwell's equation in empty space and without charges and currents, we can write the two following equations: $\bar \nabla \times \bar E= -{{\partial \bar B}\over{\partial t}}$ $\...
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When the direction of a movement changes, is the object at rest at some time?

The question I asked was disputed amongst XVIIe century physicists (at least before the invention of calculus). Reference: Spinoza, Principles of Descartes' philosophy ( Part II: Descartes' Physics, ...
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100 views

Relationship Between Differentiation in Two Frames

In section 10.3 of Principles of Ideal-Fluid Aerodynamics by Karamcheti, he writes the following: $\qquad$ Denote by $K_1$ a reference frame fixed with respect to the moving body. We shall denote ...
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42 views

Covariant derivative contracted with a metric

I would like to calculate $\nabla_\mu(g^{\mu\alpha}g^{\nu\beta}\nabla_\alpha \kappa_\beta)$. How would this expand? Where $\nabla$ is the covariant derivative, g the metric and $\kappa_\beta$ a 1-...
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Product rule of variations

I am deriving the Einstein equation using the Einstein-Hilbert action: It is obvious that the variation in the Riemann Tensor is calculated from a variational product rule. What is not obvious to ...
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1answer
61 views

Form of the Lagrangian for 1D String Dynamics

I am currently reading part of Franz Gross' Relativistic Quantum Mechanics and Field Theory, and am getting stuck on a particular point in chapter 1 where he attempts to motivate field theory by ...
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1answer
51 views

The differential of a quantity

I often see the differentials of the electric field strength and the acceleration due to gravity being written as: $$dE= \mathcal{k}\frac{dQ}{r^2} \tag{1}$$ and $$dg=\frac{GdM}{r^2} \tag{2}$$ ...
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Derivatives of polar coordiantes?

I'm a undergraduate and I was reading about the polar coordinate system specifically this paper. I don't understand the term: $$\frac{de_r}{d\theta} = e_\theta \text{, and } \frac{de_\theta}{d\theta}...
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What does it mean “differentiation with respect to the coordinates of particle 1 or 2”?

I was reading Introduction to Quantum Mechanics by Griffiths. In Chapter 5, Identical Particles, I came across the notation $\nabla_1$ and $\nabla_2$. Griffiths writes that it means "differentiation ...
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Deriving forces transformation for special relativity using the four-vector energy-momentum

I don't understand how we derive the forces transformation using the four-vector "energy-momentum". Supposing that we have two inertial frame of reference $R := \{x,y,z\}$ and $R' := \{x',y',z'\}$ ...
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90 views

Covector basis derivation

On page 65 of Schutz's A first course in General Relativity, he introduces the notation $\phi_{,\alpha}=\partial\phi/\partial x^\alpha$. He then says that $x^\alpha_{\ \ ,\beta}=\delta^\alpha _{\ \ \ \...
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Time derivative of the Lagrangian

I have the time derivative of the lagrangian: $$\frac{\mathrm d \mathcal L}{\mathrm d t}=\sum_i\left(\frac{\partial \mathcal L}{\partial q_i}\frac{\mathrm d q_i}{\mathrm d t}+\frac{\partial \mathcal ...
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1answer
87 views

What is infinitesimal displacement? [duplicate]

This section is from the Openstax University Physics: Volume 1 online textbook. In physics, work is done on an object when energy is transferred to the object. In other words, work is done when a ...
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Acceleration of speed of light [closed]

If we consider the equation "$E=mc^2$" and if we differentiate the equation with respect to time, e.g. $$\frac{\mathrm dE}{\mathrm dt}=m\frac{\mathrm dc^2}{\mathrm dt},$$ we will get after ...
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Differentiation of metric tensor in new coordinate

I want to understand the explicit meaning of $g_{\mu'\nu',\lambda}=0$ where unprimed coordinates are coordinates of the the original coordinate systems and primed ones are for new coordinate system. ...