# 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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### 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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### 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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### 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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$$\frac{\mathrm{d}}{\mathrm{d}t} \left ( \frac {\partial L}{\partial \dot{q}_j} \right ) = \frac {\partial L}{\partial q_j}.$$ 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 ...
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. ...