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6 votes
2 answers
369 views

Derivative of Delta fuction? [closed]

In Shankar's Quantum Mechanics book p-64 the last equation reads: $$ \delta'(x'-x) = -\delta'(x-x'); $$ I am confused because if I think of it using the gaussian approximation then since: $$ g(x' -x) ...
1elektron's user avatar
  • 103
0 votes
0 answers
109 views

Dirac spinor and derivative

I have question about four-derivative on spinors. $$ \bar{\psi}\partial_\mu $$ Does the derivative act on the spinor psi bar? $$ (\bar{\psi}\partial_\mu)\psi $$ $$ \bar{\psi}\partial_\mu\psi $$ Is it ...
lalala's user avatar
  • 39
0 votes
2 answers
123 views

Ambiguity in Notation for Operators in Quantum Mechanics

Let's say I am trying to find the commutator of operators $\mathbf{A}$ and $\mathbf{B}$, and I get $$[\mathbf{A},\mathbf{B}]=\nabla^2 f(x,y,z).\tag{0}$$ There seems to be some ambiguity here. In ...
Just Some Old Man's user avatar
2 votes
1 answer
1k views

What does a Umlaut (double dot) above an angle mean?

I'm reading a paper on double pendulums and there is an equation of motion that contains a double dot (Umlaut) above an angle. What does this mean / is this a standard notation in equations of motion?...
AJP's user avatar
  • 287
0 votes
0 answers
80 views

Why is cancellation of differnetial not allowed here?

This is about cancelation of differentials .I am learning basics of tesnor from "Mathematical Methods " by Boas. There I encountered this epression which author says are equal. $$ \frac{\...
mum's user avatar
  • 128
7 votes
3 answers
1k views

Vector cross product formula without a second term (Spiegel, Theoretical Mechanics)

In Spiegel's Outline Of Theoretical Mechanics (more precisely in the Moving Coordinate Systems chapter, § "Derivative Operators") I find (both in the 1968 and the 1977 edition) the following ...
Vince Vickler's user avatar
0 votes
0 answers
138 views

Mathematical definition and notation of Fermat's Principle for least time

I was going through geometrical optics as it's a part of my undergrad course ,and I found about the Fermat's principle. The principle was understood but the mathematical equation given for it was not ...
PATRICK's user avatar
  • 345
2 votes
1 answer
115 views

Schrödinger equation: $\frac{\partial}{\partial t}$ and $\frac{d}{dt}$ [duplicate]

I have seen two different forms of Schrödinger equation: $$i\hbar\frac{\partial|\psi(t)\rangle}{\partial t}=\hat{H}|\psi(t)\rangle$$ and $$i\hbar\frac{d|\psi(t)\rangle}{d t}=\hat{H}|\psi(t)\rangle.$$ ...
TaeNyFan's user avatar
  • 4,276
-1 votes
1 answer
650 views

What do some of the symbols in the Schrodinger Equation mean? [closed]

The Time Dependent Schrodinger Equation has the form $$i\hbar\frac{\partial}{\partial{t}}\Psi=-\frac{\hbar^2}{2m}\left(\nabla^2+V\right)\Psi$$ and the Time Independent Schrodinger Equation has the ...
Anders Gustafson's user avatar
12 votes
6 answers
3k views

Using differentials in physics [duplicate]

I was lately wondering about the use of differentials in physics. I mean, usually $dx$ is thought of as a small increment in $x$, but does this have any rigorous meaning mathematically. Doubts started ...
Gaston Castillo's user avatar
1 vote
4 answers
420 views

What do $\nabla$ and $\frac{d }{d t}$ mean when they are by themselves?

In QM and QFT, I have seen some equations where they have just the derivative and/or the gradient without specifying what it is acting on. Taken from wiki. This does not make sense to me since I ...
Tachyon's user avatar
  • 2,042
7 votes
2 answers
652 views

Issue in deriving Ehrenfest's theorem

Working in Schrodinger picture, while deriving Ehrenfest's theorem, we go - $$ \frac{d}{d t}\langle A\rangle=\frac{d}{d t}\langle\psi|\hat{A}| \psi\rangle $$ $A$ is an operator. Expanding RHS- $$ \...
aneet kumar's user avatar
0 votes
0 answers
108 views

What does $\nabla'$ mean? [duplicate]

In D.J Griffiths Electrodynamics (Page 173) it says, $\nabla' |\vec{x}| = \frac{\hat{x}}{x^2}$. However by my calculation $\nabla |\vec{x}| = -\frac{\hat{x}}{x^2}$ so what does the $\nabla'$ signify?
Rodrigo Guevarez's user avatar
0 votes
2 answers
204 views

Physical meaning of the exterior derivative of the first law of thermodynamics

We know, $$ dU = d \overline{q} - d \overline{W}.$$ suppose we took the exterior derivative on both sides, then: $$ 0= d( d \overline{q}) - d( d \overline{W})$$ This means, $$ d^2 \overline{q} = d^2 \...
Brian's user avatar
  • 8,040
0 votes
1 answer
209 views

Tensor notation of covariant derivative

I'm trying to apply Wald's General Relativity equation $3.1.14$: $$\nabla_a{T^{b_1\dots b_k}}_{c_1\dots c_{\ell}}=\overline{\nabla}_a{T^{b_1\dots b_k}}_{c_1\dots c_{\ell}}+\sum_i{C^{b_i}}_{ad}{T^{b_1\...
FonerBalear's user avatar
0 votes
1 answer
34 views

Help decipher the notation said to denote a common pattern in various branches of science in Prelude to Mathematics by W. W. Sawyer

In Section 1.2 - Nature's Favorite Pattern? (excerpted below) of Prelude to Mathematics by W. W. Sawyer (1982), he said mathematicians used the notation $\nabla^2 V$ to denote a pattern that occurs &...
reflectionalist's user avatar
0 votes
2 answers
326 views

Is there any difference in superscript and subscript notation in finite difference method

Is there any difference in superscript and subscript notation in the finite difference method? I see the same paper use (superscript for $x$ and superscript for $y$ notation) and (subscript for x and ...
Abinash's user avatar
2 votes
3 answers
109 views

Small doubt on derivatives acting on kets/bras

I have a quick, silly question. If $\psi(x):=\langle x|\psi\rangle$, does the bra $\langle x|$ 'go through' the $\partial_x$ operator, as in $$\langle x|\partial_x|\psi\rangle=\partial_x\psi(x) \quad ?...
Brown Hole's user avatar
3 votes
2 answers
93 views

What is the meaning of the equation of the change in entropy? [duplicate]

In my chemistry book, the formula for change in entropy is given as : $$\int{dS} = \int{\frac{δq_{rev}}{T}}$$ What is the meaning of $δq_{rev}$? I know that it is the heat exchanged in a reversible ...
RIPAN BARUAH's user avatar
2 votes
1 answer
87 views

How to express the elementary work definition as an explicit functional expression [duplicate]

My assumption here is that in the definition of elementary work : $dW = F ds$ symbol $d$ represents a differential. But a differential implies a function : $dy =_{df} d[f(x)] = f'(x) \Delta x = f'(...
Floridus Floridi's user avatar
0 votes
1 answer
74 views

Analogous notation to $\nabla$ but for gradient with respect to $\vec{k}$ not $\vec{x}$

$\nabla = \frac{\partial}{\partial x_i}$ so $\nabla F = (\frac{\partial F}{\partial x}, \frac{\partial F}{\partial y}, \frac{\partial F}{\partial z})$. However, is there a similar equalivalent notion ...
Alex Gower's user avatar
  • 2,654
42 votes
3 answers
4k views

Partial derivative notation in thermodynamics

Most thermodynamics textbooks introduce a notation for partial derivatives that seems redundant to students who have already studied multivariable calculus. Moreover, the authors do not dwell on the ...
1__'s user avatar
  • 1,634
2 votes
3 answers
266 views

Is the equation $[\nabla_{\mu},\nabla_{\nu}]=F_{\mu\nu}$ correct? If yes, how does it have to be interpreted?

It seems like simply using the equation \begin{equation} \nabla_{\mu}=\partial_{\mu}+A_{\mu} \end{equation} isn't enough: One obtains \begin{equation} [\nabla_{\mu},\nabla_{\nu}]=\underbrace{[\...
Filippo's user avatar
  • 1,911
0 votes
1 answer
132 views

Vector calculus in Electromagnetism [closed]

I found a problem which had $$\partial_i (A_j \vec{G})= (\vec{\nabla} .\vec{ A} )\vec{G}+ (\vec{A}.\nabla) \vec{G} $$ but my problem is what does $$\partial_i (A_j \vec{B})$$ even mean? it doesn't ...
SHIN101's user avatar
  • 63
2 votes
2 answers
172 views

Conjugate momentum notation

I was reading Peter Mann's Lagrangian & Hamiltonian Dynamics, and I found this equation (page 115): $$p_i := \frac{\partial L}{\partial \dot{q}^i}$$ where L is the Lagrangian. I understand this is ...
math-ingenue 's user avatar
0 votes
1 answer
305 views

What does an "elementary value $\delta$ of a quantity" mean?

In page-11 of I.E irodov Fundamental laws of mechanics, some notation used in the book is introduced. There, it is said that $\delta$ denotes the elementary value of a quantity but what exactly does ...
Brian's user avatar
  • 8,040
3 votes
5 answers
219 views

Confusion about $\partial_\mu x^\mu = 4$

Why is it that $\partial_\mu x^\mu = 4$? I thought that $\partial_\mu x^\mu$ could be expanded as $$\partial_\mu x^\mu = -\partial_1x^1 + \partial_2x^2 + \partial_3x^3 + \partial_4x^4 \\ =-1+1+1+1\\ =...
Lazarus's user avatar
  • 165
3 votes
0 answers
546 views

Commutation relation of four vectors [closed]

I was trying to prove that: $$[P_\mu, J_{\rho \sigma}] = i(\eta_{\mu \sigma} P_\rho - \eta_{\mu \rho} P_\sigma) $$ $\textbf{Attempt}$ $$\begin{align} [P_\mu, J_{\rho \sigma}] = [P_\mu, x_\rho P_\sigma ...
RKerr's user avatar
  • 1,327
5 votes
4 answers
307 views

Newton's Law of Cooling: $\delta Q$ or $\mathrm{d}Q$?

In this popular answer, I invoked Newton's Law of Cooling/Heating: $$\dot{q}=hA\Delta T\tag{1}$$ $$\dot{q}=\frac{\mathrm{d} Q}{\mathrm{d}t}\tag{2}$$ $$\dot{q}=\frac{\delta Q}{\mathrm{d}t}\tag{3}$$ $$\...
Gert's user avatar
  • 35.5k
1 vote
2 answers
353 views

Are there differences in notation for the d'Alembert operator?

On Wikipedia the d'Alembert operator is defined as $$\square = \partial ^\alpha \partial_\alpha = \frac{1}{c^2} \frac{\partial^2}{\partial t^2}-\nabla^2 $$ However, my professor uses the notation: $$ \...
Struggling_Student's user avatar
4 votes
1 answer
111 views

What does $\mathbf{A}\cdot\nabla$ mean here?

What does $\mathbf{A}\cdot\nabla$ mean in an expression like $(\mathbf{A}\cdot\nabla)\mathbf B$? I found this in Griffiths’ Classical Electrodynamics book and cannot figure it out.
Pranshu Khare's user avatar
3 votes
2 answers
133 views

Is this notation inconsistent? If not, can some explain why not?

Im working through a textbook section on particle kinematics. An example given is relating vertical velocity to horizontal velocity and states: $y$ has a constant velocity of $10 \ \rm [m/s]$ $y=(0....
RoRo's user avatar
  • 31
1 vote
2 answers
701 views

Infinitesimal Changes - Notations

in my thermodynamics class we saw the following formulas: $$ dS = \frac{\delta Q}{T} $$ and $$ \delta W = PdV $$ This was part of a review of thermodynamics that we have seen previously; however, in ...
STOI's user avatar
  • 348
1 vote
1 answer
101 views

What does the $d$ mean in metric tensor calculations?

In many metric calculations, like the Schwartzschild metric, we see formulas like $d^2X / dt^2$ and many other formulas with a $d$ in them. You'd be surprised that I've been looking for months to ...
foolishmuse's user avatar
  • 4,855
1 vote
2 answers
305 views

Is $ \partial_{\mu} \partial^{\mu} $ the second derivative or derivative squared?

This might be a silly question, but I'm just getting my feet wet with field theories. So far I have assumed that $ \partial_{\mu} \Phi\partial^{\mu}\Phi $ means $ (\Phi_t)^2-(\Phi_x)^2-...$ . I ...
Johnny's user avatar
  • 163
0 votes
3 answers
580 views

Meaning of the notation $\sigma_{ji,j}$

In page 28 of the book Introduction to Linear Elasticity, 4ed by Phillip L. Gould · Yuan Feng, it says $$ \int_V{\left( f_i+\sigma _{ji,j} \right) \text{d}V=0} $$ What does it mean by writing $\sigma ...
FFjet's user avatar
  • 103
8 votes
1 answer
999 views

Two different versions of Schrödinger's equation - are they equivalent?

For simplicity, let's look at the case of one particle in one dimension. We usually think of the wave function as a function \begin{align} \Psi\colon\mathbb R\times[0,\infty[&\to\mathbb C\\ (x,t)&...
Filippo's user avatar
  • 1,911
0 votes
1 answer
1k views

Commutator of covariant derivative and field $F_{\mu \nu}$

I am working with the covariant derivative and trying to show that the commutator of this derivative $[D_\mu , D_\nu]$ is proportional to the field $F_{\mu \nu}$. That is, I need the final term to be ...
user avatar
1 vote
3 answers
368 views

Commutation relation of $e^{ikx}$ and $\partial_x$ in Nakahara

I'm reading through Nakahara's Geometry, Topology and Physics and I don't understand the following derivation on pg. 41: $$ \text{Now we find from the commutation relation of } \partial_x \equiv \frac{...
Feng's user avatar
  • 432
2 votes
3 answers
193 views

Is $ d \mathbf v · d \mathbf v = d \mathit v^2 $?

My teacher has proved the following: $$ \mathit v^2 = \mathbf v·\mathbf v = \frac{d\mathbf r}{dt}·\frac{d\mathbf r}{dt} = \left(\frac {ds}{dt}\right)^2 \Rightarrow \mathit v = \frac{ds}{dt} $$ Because ...
Pascu22's user avatar
  • 23
0 votes
1 answer
1k views

What is $D$ or $D$-with-a-slash-through-it in the Standard Model equation(s)?

In the mathematical formulation of the Standard Model, which I do not understand yet, there is a capital letter $D$ or $D$-with-a-slash-through-it that I can't find an explanation for. Flip Tanedo (a ...
Kurt Hikes's user avatar
  • 4,709
-1 votes
2 answers
603 views

What does $d$ stand for in this formula?

Context: I am building a tennis ball machine and am having trouble interpreting the following formula for the flight path of the ball. I know all of the other values in the formula but the source I am ...
ShinyWhaleFood's user avatar
14 votes
3 answers
1k views

What is meant by a partial derivative of a ket?

In my QM book I often see partial derivatives mixed with kets, like $$ \frac{\partial}{\partial a} |\psi \rangle $$ where $a \in \{x, y, z\}$. Here I'm assuming that $| \psi \rangle \in \mathbb{C}^n$ ...
George's user avatar
  • 337
3 votes
3 answers
1k views

Navier Stokes: $(u⋅∇)u$ vs $u⋅∇u$

I can find this term stated both ways in different literature. Are they equivalent? It's weird because the dot is a dot product in (u⋅∇), but ∇u being a gradient of a vector field, would (presumably) ...
TravisG's user avatar
  • 339
0 votes
1 answer
110 views

Computing derivatives "at constant" quantities in thermodynamics

What does it mean in thermodynamics when a derivative is computed "at constant $X$"? If I see $\left.\frac{\partial S(E, N)}{\partial E}\middle| \right._N$ how is the derivation performed ...
pretzlstyle's user avatar
1 vote
1 answer
77 views

Interpretation of Variation Notes

I would like an explanation to how this Lagragian partial derivative was taken (eq. 3). This probably is more suited for the math Stack Exchange, however this is for a physics course which is why I am ...
Tom's user avatar
  • 15
3 votes
0 answers
66 views

How a 'variation' $\delta x$ of an independent parameter differs from $dx$? [closed]

I have been reading the Classical Field theory part from The Quantum field theory book of Lewis H Ryder. After defining classical field $\phi(x^\mu)$ he says something about adding variations on both ...
Baibhab Bose's user avatar
3 votes
1 answer
1k views

What does $\overset\leftrightarrow{\partial_{\mu}}$ means?

I have a scalar complex field: $\phi(x) = \phi_{1} + i \phi_{2}\;$ so $\;\phi^{*}(x) = \phi_{1} - i \phi_{2}$ where $\phi_{1}, \; \phi_{2}$ are real scalar fields. Then I have something like $\;\phi^{...
amith's user avatar
  • 83
1 vote
1 answer
830 views

How Total derivative got converted into partial derivative?

While studying the book Heat and Thermodynamics by Zemansky and RH Dittman, in the topic 'equation for a hydrostatic system' (page no. 88) it was given in equation 4.12, when we take Pressure P ...
Kaushalesh Upadhyay's user avatar
0 votes
2 answers
2k views

Partial derivative of the function with respect to $t$ in total derivative

In the formula description there is one extra partial derivative compared to the example solution. What's the difference here? What's the physical implication of the last partial derivative in the ...
Ali's user avatar
  • 53