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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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41 views

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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45 views

Why the continuous arrangement of point masses (particles) at infinitesimal separations leads to a extended system?

I am basically talking in terms of Newtonian mechanics. The Newton's laws started with a good and easy assumption of particles as point masses. This assumption clearly reformed physics and a great ...
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2answers
48 views

Why do we differentiate displacement to get velocity if displacement is given as a function of time [on hold]

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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57 views

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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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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52 views

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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28 views

Physics exercises with solutions that use derivatives and integrals [on hold]

I'm looking for physics exercises, mainly from classical mechanics and power, work, energy, that focus on creating a mathematical model using differentiation and integration to obtain the solution. ...
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1answer
92 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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34 views

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
36 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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173 views

Convective derivative vs total derivative

I was wondering what is the difference between the convective/material derivative and the total derivative. We were introduced to the notion of material derivative $$ \frac{D\vec{u}}{Dt}=\frac{\...
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1answer
53 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
47 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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What does the metric condition $\nabla_\rho g_{\mu\nu}=0$ in General Relativity intuitively mean for an observer measuring distances?

In General Relativity, the following condition hold: $\nabla_\rho g_{\mu\nu}=0$, where $g_{\mu\nu}$ is the metric of spacetime which has to do with measuring distances and angles and $\nabla$ is the ...
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2answers
64 views

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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1answer
234 views

exponential autocorrelation function by approximation of derivative

I have been pondering about the following question: Given a time-dependent function $f(t)$, is it possible to show that its autocorrelation function will generally follow a decaying exponential ...
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62 views

How to further expand $\text{grad} \left( \vec{a} \cdot\vec{b} \right ) = \vec{\nabla} \left (\vec{a} \cdot\vec{b} \right )$? [migrated]

With $\vec{a}, \vec{b}: \mathbb{R}^3 \to \mathbb{R}^3$ vector fields: I want to expand $\text{grad} \left( \vec{a} \cdot \vec{b} \right ) = \vec{\nabla} \left (\vec{a} \cdot \vec{b} \right )$. So I ...
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3answers
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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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45 views

Writing del, divergence, and curl in generalized coordinates [migrated]

In three dimensional Cartesian coordinates the Hamilton operator, del, is written as $\nabla= \begin{pmatrix} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} ...
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87 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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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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50 views

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
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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 ...
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How to differentiate an operator in QM?

I recently started learning QFT and the lecturer wrote down some equations for a quantum harmonic oscillator: $$\hat H= \frac{\hat p^2}{2m}+\frac{1}{2}mw^2\hat x^2$$ $$\partial^2_t \hat x=-w^2\hat x$$ ...
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1answer
85 views

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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Derivation of Rotational Motion Equations using Calculus

How are the equations for rotational motion derived using calculus and the following general equations ? $$\mathbf{v}(t) = \mathbf{v}_0+\int_{t_0}^t \mathbf{a}(t')dt'$$ $$\mathbf{r}(t) = \mathbf{r}...
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58 views

Differentiation of the determinant $g$

Let $g$ be the determinant of the metric tensor. I want to derive the following equation $g_{,\nu}=gg^{\lambda \mu}g_{\lambda \mu,\nu}$. It is said that $gg^{\lambda \mu}$ is a cofactor, but I can't ...
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74 views

Converting velocity vector formula from Cartesian coordinate system to polar coordinate system

I have a little question about converting Velocity formula that is derived as, $$\vec{V}=\frac{d\vec{r}}{dt}=\frac{dx}{dt}\hat{x}+\frac{dy}{dt}\hat{y}+\frac{dz}{dt}\hat{z}$$ in Cartesian Coordinate ...
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27 views

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. ...
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1answer
1k views

Physical significance of divergence

In my textbook They considered a parallelopiped $ABCDEFGH$ with sides $dx,dy,dz$ parallel to $x,y,z$ axis respectively $\vec V$ represents the vector velocity of the fluid at the centre $P$ of f ...
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49 views

The two definitions of the divergence of a vector field? [migrated]

Now, I am aware that the divergence of a vector field, $\vec{F}$, can be defined in two ways. What I don't understand is why do these equal each other formally? Definition 1: $$\text{div}\vec{F} = \...
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28 views

Delta over variable relations

In the text I'm reading, which discuss the heat transfer by convection in stars, it shows that the relation $\rho \propto \frac{P}{T}$ (equation of state for an ideal gas) implies $$\frac{\Delta\rho}{\...
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1answer
60 views

How to take derivative with respect to Lagrangian of complex field?

Basics: The Lagrangian in field theory was written as $$\frac{\partial \mathfrak{L}}{\partial \varphi}=\partial_\mu(\frac{\partial\mathfrak{L}}{\partial(\partial_\mu\varphi)})$$. Question 1: Is $\...
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47 views

How does vectorization affect $\nabla$?

The homework and exercise was to prove $\nabla \times {A}$ transform as a vector, and I've solved it thorough hard algebra. However, something occurred to my mind and I have a hard time to resolve it....
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1answer
51 views

(Physics version of) Taylor expansion. In the the context of deriving a Lie groups generators (a Lie algebra from a Lie group)

Statement which I'm confused about: "Consider some n-dimensional Lie group whose elements depend on a set of parameters $\alpha = (\alpha_1 ... \alpha_n)$, such that $g(0) = e$ with e as the identity,...
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3answers
101 views

If $\mathrm df$ is an inexact differential, how would the function $f$ look like?

I am studying thermodynamics and in the first chapter the concept of exact and inexact differentials were used to talk about the differences between internal energy, work and heat. From Blundell and ...
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3answers
111 views

Proof for Vector Identity

I am currently studying electrodynamic and came across the following vectoridentity, but I am unable to prove it: $$ \vec{f} \times ( \nabla \times \vec{f} ) -\vec{f}(\nabla\cdot\vec{f}) = \nabla \...
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1answer
66 views

Find $v(t)$ and $x(t)$, How do I treat $δt$? [closed]

We apply a force to a particle with a mass $m$ and inicial velocity $v_0$: $$ F(t) = \left \{ \begin{matrix} 0 & \mbox{ $t<t_0$} \\ \frac{p_0}{\delta t} & \mbox{ $t_0<t<t_0 +\...
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33 views

Fractional differential equations and Physics [duplicate]

Are the "fractional differential equations" have any real significance in respect to physics? or are they just stilted math?
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1answer
67 views

Divergence of a vector multiplied by dot product [closed]

If I am correct, then $\operatorname{div} [(\vec A\cdot \vec B)\vec C] = (\vec A \cdot \vec B) \operatorname{div} \vec C + \vec C \cdot \nabla (\vec A\cdot\vec B)= (\vec A \cdot \vec B) \operatorname{...
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1answer
102 views

What was the real need of the operators of divergence and curl? [closed]

As I'm advancing my study in Electromagnetism I'm getting introduced to more mathematical operators which are exclusively used in Electromagnetism and Fluid Dynamics only. Let me try to explain ...
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68 views

Derive ballistic equation using conservation of energy, stuck with (dx/dt)^2 [closed]

When observing a falling particle, it's easy to derive the trajectory using the following reasoning: the particle is subject to one external force mg and by taking ...
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1answer
37 views

Meaning of normal acceleration?

acceleration means the rate of change in velocity (vector quantity) and the differentiation means to divide a certain quantity into small elements (i.e $dx$) as we do to find the acceleration at any ...
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4answers
106 views

Why the need for defining the velocity as a derivative? [closed]

Something intuitive and fundamental as the concept of velocity (of a particle for example) in classical physics is defined as a derivative, a concept to me quite vague and strange, although i know its ...
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56 views

How is this equality true? Lagrangian mechanics

So this equality was given as a part of how to derive the Lagrangian equality from Newton's second law. How it is true? $$\frac{d\pmb{p}_k}{dt}\cdot\frac{\partial\pmb{r}_k}{\partial q_i} = \frac{d}{...
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2answers
63 views

Leibniz Rule for Covariant derivatives

I recently came across a video by prof Fredrick Schuller on general relativity where he defines the leibniz rule to be, $\nabla_X (T(\omega,Y))=\nabla_XT(\omega,Y)+T(\nabla_{X} \omega,Y)+T(\omega,\...
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1answer
131 views

Statistical physics is unable to prove that $TdS=d\overline{E}$

I will pose $k_B=1$. Suppose a system of statistical physics with the constraints: $$ \begin{align} 1&=\sum_{q\in\mathbb{Q}}\rho(q)\\ \overline{E}(\beta)&=\sum_{q\in\mathbb{Q}} E(q)\exp(-\...
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81 views

Is rate of change of internal energy with respect to volume at constant temperature always zero?

According to substitutions of some equations in "Heat and Thermodnamics by Zemansky", $\left(\frac{\partial U}{\partial V}\right)_T$ turns out to be zero. First $T\ dS$ relation is: $$TdS=C_v\ dT+T\...
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1answer
50 views

Why do we neglect $\Delta t^2(\frac{d\vec{r}}{dt}\frac{d\vec{\hat{r}}}{dt})$ at Taylor Expansion?

I'm just started to Ankara University Physics Department two weeks ago. I have missed my 2 hours of PHY105 course that is the last week Wednesdey. The subject that i missed was Derivatives of Vectors. ...
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1answer
30 views

Do Equal Time Commutation / Anticommutation relations automatically also apply for spatial derivatives?

The question is basically in the title. My naive thought was that when a commutation relation holds for all field operators $\Psi(\vec{x})$ (by "all" I mean "at all positions $\vec{x}$") on a fixed ...