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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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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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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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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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Worked examples of Lie derivative [migrated]

I'm trying to find the Lie derivative of a 2 form $\sin(\theta)d\theta d\phi$ with respect to a vector field given in a differential basis and I think the way to go here is to use Cartan's formula but ...
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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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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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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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Kinematics lessons involving calculus [closed]

Does solving instantaneous velocity involves calculus in kinematics? What are the other lessons or unknown variables in Kinematics that involves the use of Calculus? What are the other lessons also or ...
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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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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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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 & -...
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General Relativity: Exchanging a field with its infinitesimal components on metric tensor

On a youtube video about Einstein's field equations, the author writes the following equation (https://youtu.be/foRPKAKZWx8?t=1078): $$d\phi=\sum_{n} \frac{\partial \phi}{\partial x^n} dx^n\tag{1}$$ ...
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What is the physical meaning of divergence? [duplicate]

I want to visualize the concept of divergence of a vector field. I also have searched the web.Some says it is 1.the amount of flux per unit volume in a region around some point 2.Divergence of ...
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What does $\delta$ represents in FLUCTUATION-DISSIPATION THEOREM?

i am trying to follow the following tutorial. I keep seeing $\delta$ over functions such as $\delta F(x)=F(x)-\langle F(x)\rangle_t$ (Eq 14.4) in this and in other tutorials and questions here. What ...
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Do we need really need tetrads? OR: Does the No-Go theorem for spinors in curved space only apply to a linear connection?

When I first learned General Relativity, the tetrad formalism was introduced with near simultaneity. I was immediately taught that, to utilize spinors in any way, I had to formulate a local ...
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What is the time derivative $\frac{d}{dt}(\exp(\hat A))$ of operator exponential $\exp(\hat{A})$? [duplicate]

Assuming that the operator $\hat{A}$ does not necessarily commute with $d \hat A/dt$, what is $$\frac{d}{dt}(\exp(\hat A))~?\tag{1}$$ Is it $\exp(\hat A(t)) \frac{d \hat A}{dt}$ or $\frac {d\hat A}{dt}...
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How can you explain $cot(\alpha)= \dfrac{d}{d\theta}\cdot ln(r) = \dfrac{1}{r} \dfrac{dr}{dt}$ in a polar coordinate system? [migrated]

This alinea is about the $$cot(\alpha)= \dfrac{d}{d\theta}\cdot ln(r) = \dfrac{1}{r} \dfrac{dr}{dt}.$$ Where does the $ln (r)$ come from? How can you derivate it from that picture? I want to use that ...
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When can we raise lower indices on “nontensors” as described in Dirac's book *General Theory of Relativity*?

This is a follow on to my previous question: Dirac's book *General Theory of Relativity*: Doesn't this show the partial derivative of the metric tensor is zero? I should not have made that ...
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1answer
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What is the industry standard definition of $\nabla\cdot\mathfrak{T}$ (del dot a tensor)? Re: MTW

In chapter 3 of MTW's Gravitation using the example of a rank-3 tensor $\mathfrak{S}$ they define $$\text{(divergence of }\mathfrak{S}\text{ on the first slot)}\equiv{\nabla\cdot\mathfrak{S}}$$ ...
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Why an elementary work is written $\delta W$ instead of $dW$? [duplicate]

Why an elementary work is written $\delta W$ instead of $dW$? For example, it's often written $$\delta W=F\cdot dr$$ if $dr$ is the elementary displacement. Why don't we write as usual $dW=F\cdot dr$?
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Dirac's book *General Theory of Relativity*: Doesn't this show the partial derivative of the metric tensor is zero?

See the bold text for my question. This question regards Dirac's General Theory of Relativity page 13. In his demonstration that the length of a vector is unchanged by parallel displacement Dirac ...
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1answer
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Why $(\frac{\partial S}{\partial T})_P=(\frac{\partial S}{\partial T})_V+(\frac{\partial S}{\partial V})_T(\frac{\partial V}{\partial T})_P$?

In the thermodynamics book (Adkins) I'm using, the following relation is cited without reference but I am not sure where it comes from. $$\left(\frac{\partial S}{\partial T}\right)_P=\left(\frac{\...
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Tensorality of the Lie Derivative and $i([X,Y]) = [L(X),i(Y)]$

I'm trying to understand the equation following (2.15) on p.9 of Blau's Symplectic Geometry and Geometric Quantization. For two vector fields $X,Y$ on a symplectic manifold $M$ we are told one ...
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1answer
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Ambiguous curl for a vector field

I'm trying to work out how to explain why the curl/divergence for the following image is ambiguous without giving the explicit vector function for the following: E has an ambiguous divergence and I ...
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Thermodynamics and differential forms

In Potter's Thermodynamics: Demystified (page 68), the author wrote: Since only two independent variables are necessary to establish the state of a simple system, the specific internal energy can ...
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1answer
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How come $\frac{d}{dt}\left(\frac{\partial {r_i}}{\partial {q_j}}\right) = \frac{\partial {\dot r_i}}{\partial {q_j}}$ in Lagrangian mechanics? [duplicate]

It is written in the Goldstein's Classical Mechanics text that $$\frac{\mathrm d}{\mathrm dt}\left(\frac{\partial {r_i}}{\partial {q_j}}\right) = \frac{\partial {\dot r_i}}{\partial {q_j}}=\sum_k \...
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1answer
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Why aren't Christoffel symbols tensors? - asked from a fibre bundle perspective

I've been reading about connections on fibre bundles recently and it's made me think about the exact nature of the Christoffel symbols in GR. If we have a vector bundle $E$ over $M$ and put a ...
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1answer
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Why is $\partial_{\mu}x^{\nu} = \delta^{\nu}_{\mu}$?

In Blundell's book on QFT, one can find the following Is this because of: $$\partial_{\mu}x^{\nu} = \partial_{\mu}x^{\nu^{'}} \partial_{\nu^{'}}x^{\mu}$$ $$\partial_{\mu}x^{\nu} = \Lambda_{\mu}^{\...
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How do “divergence” and “curl” relate to the three states of matter?

A fluid is said to have divergence (the ability to flow) and curl {the ability to rotate). Do these two characteristics fully define a fluid, or are there other important properties that I am missing? ...
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Question about derivation of four-velocity vector

In order to describe a notion of rate of change of positon, in four-dimensional spacetime, we have to introduce the concept of four-velocity. So, consider the following: For a massive particle ...
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Physical Interpretation of d'Alembert Operator

$$\mathop{{}\Box}\nolimits=\frac{1}{c^2}\frac{\partial^2}{\partial t^2}-\mathop{{}\bigtriangleup}\nolimits$$ is the d'Alembert-operator. It seems to consist of an oscillation and a diffusion. Is there ...
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1answer
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Double divergence of stress tensor for migration flux

I am looking to calculate migration as a function of time using equation in Image 1. SigmaP is the total particle stress tensor in the cylindrical coordinates (r, theta, z). I am only interested in ...
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1answer
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Commute covariant derivatives of spinors

Consider a spinor field $\psi$ on a general smooth Lorentzian manifold. Let $\Sigma_{ab}$ be a matrix representation of the Lorentz group, and let Greek/Latin letters represent world/Lorentz indices. ...
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Confused with derivative and partial derivative

suppose $x=f(t)$ with a constant acceleration. Then does $\frac{\text d x}{\text d t} = \frac{\partial (x)}{\partial(t)}$ since the position in $x$ only changes with time? Then the acceleration in ...
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1answer
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Definition of integral functional [duplicate]

I'm reading the section of Marion and Thornton devoted to basics on the Calculus of Variations, and came across this definition for the functional: $$J = \int f(y(x), y'(x);x) dx$$ implying that $f$ ...
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1answer
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Why is velocity mathematically describes as a division? [duplicate]

I want to know why, in kinematics, is velocity described as $v = \frac{\Delta x}{\Delta t}$, and why it is not described as any other expression (like a multiplication), why does a division is the one ...
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Theoretical definition and pratical mesurement of differential cross section

In Sakurai's book, the definition of differential cross section is: $$d\sigma/d\Omega= transition \;rate / probability\; flux $$ However this def doesn't contain any information about the thickness ...
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1answer
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Relation between curl and gradient [duplicate]

I need to prove the following relation (with vector $\mathbf{V}$) : $$(\mathbf{V} \cdot \nabla)\mathbf{V} = \frac{1}{2}\nabla (\mathbf{V}^2)+(\nabla \times \mathbf{V}) \times \mathbf{V}\quad\quad\...
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With what velocity are we moving along the time dimension?

Does the question make sense? Velocity along time axis means $v_t=\mathrm dt/\mathrm dt$? If it doesn't, please explain where the flaw is. Taking time as measure like length? Or do we need to ...
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1answer
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Vector Identity For Electrostatics

I am reading about electrostatics and came across this vector identity when discussing the $D$ field: $$\frac{\nabla k_{e}}{k_{e}} = \nabla \ln (k_{e}).$$ I have not seen this identity before and ...
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1answer
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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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Where is the error in this calculation of net curl for simple magnetic field?

I wasn't sure whether to post this on MSE, but PSE seems more appropriate. Let B be a static magnetic field in spherical coordinates, defined as $B=r\hat{\theta}$. Then, it's curl is $$\nabla \times ...
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Operator $A$ only act on the neighboured state or operator but not the entire expression?

In state vector formalism $A|\psi(x)><u(x)|=(A|\psi(x)>)<u(x)|$, where $A$ only act on $|\psi(x)>$ However, in terms of wave formalism, suppose $A$ is the well known $\frac{d}{dx}$. ...
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D'Alembert Operator

In which book or where can I find the derivation of the d'Alembert operator? \begin{equation} \Box \psi= \frac{1}{\sqrt{-g}}\partial_\mu \left( \sqrt{-g}\partial^\mu \psi \right) \end{equation}
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Is $\nabla=\nabla'$? Nabla operator acting on $r^n$

I have been taught that $$\nabla r^n =\text{gradient of }r^n =n r^{n-1}\ \hat{\boldsymbol r}$$ but in introduction to electrodynamics by Griffith (4th edition) on page 173, $\nabla' r^n$ is given by $-...
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Why are higher derivatives not important?

While doing many physics problems I have noticed people ignoring the higher derivatives of a function in an equation. This is a frequent trick used by many that the fact that I cannot understand the ...
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1answer
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Taylor expansion of scalar fields [closed]

Starting of with electrodynamics I have to compute the taylor expansion around $\vec{r} = 0$ of $\psi (\vec{r}) = |\vec{r} - \vec{r_0}|^{\frac{3}{2}}$ where $\vec{r_0}$ is a constant vector up to ...
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Have fractional order differential models been explored as an alternative to standard gravitational field theory?

Since Einstein introduced his field equations and general theory of relativity, experimental evidence, at least on the cosmic scale has repeatedly supported the theory. Nevertheless, many seeking to ...
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Planck Blackbody Radiation: Is this an error in the textbook?

the textbook I am reading describes two forms of equations of Blackbody Radiation. $$d\rho(\nu, T) = d\rho_\nu(T)d\nu = \frac{8\pi h}{c^3}\ \frac{\nu^3d\nu}{e^{h\nu/k_BT}-1}\ . $$ Substituting $ c = \...
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1answer
47 views

Dot product in cylidrical coordinates

I'm given the vector: $$\vec{V}{(r,θ,z)}=\frac{1}{r}\hat{e_r} + (r\cosθ)\hat{e_θ}+\frac{z^2}{r^2}\hat{e_z}$$ I want the scalar product ${\vec{\nabla}}\cdot{\vec{V}}$ We know that in cylindrical ...