All Questions
Tagged with differentiation notation
224 questions
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:
$$ g(x' -x) ...
0
votes
0
answers
109
views
Dirac spinor and derivative
I have question about four-derivative on spinors.
$$
\bar{\psi}\partial_\mu
$$
Does the derivative act on the spinor psi bar?
$$
(\bar{\psi}\partial_\mu)\psi
$$
$$
\bar{\psi}\partial_\mu\psi
$$
Is it ...
0
votes
2
answers
123
views
Ambiguity in Notation for Operators in Quantum Mechanics
Let's say I am trying to find the commutator of operators $\mathbf{A}$ and $\mathbf{B}$, and I get
$$[\mathbf{A},\mathbf{B}]=\nabla^2 f(x,y,z).\tag{0}$$
There seems to be some ambiguity here.
In ...
2
votes
1
answer
1k
views
What does a Umlaut (double dot) above an angle mean?
I'm reading a paper on double pendulums and there is an equation of motion that contains a double dot (Umlaut) above an angle. What does this mean / is this a standard notation in equations of motion?...
0
votes
0
answers
80
views
Why is cancellation of differnetial not allowed here?
This is about cancelation of differentials .I am learning basics of tesnor from "Mathematical Methods " by Boas. There I encountered this epression which author says are equal. $$ \frac{\...
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 ...
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 ...
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
$$i\hbar\frac{d|\psi(t)\rangle}{d t}=\hat{H}|\psi(t)\rangle.$$
...
-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 ...
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 ...
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 ...
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-
$$
\...
0
votes
0
answers
108
views
What does $\nabla'$ mean? [duplicate]
In D.J Griffiths Electrodynamics (Page 173) it says, $\nabla' |\vec{x}| = \frac{\hat{x}}{x^2}$. However by my calculation $\nabla |\vec{x}| = -\frac{\hat{x}}{x^2}$ so what does the $\nabla'$ signify?
0
votes
2
answers
204
views
Physical meaning of the exterior derivative of the first law of thermodynamics
We know, $$ 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 \...
0
votes
1
answer
209
views
Tensor notation of covariant derivative
I'm trying to apply Wald's General Relativity equation $3.1.14$:
$$\nabla_a{T^{b_1\dots b_k}}_{c_1\dots c_{\ell}}=\overline{\nabla}_a{T^{b_1\dots b_k}}_{c_1\dots c_{\ell}}+\sum_i{C^{b_i}}_{ad}{T^{b_1\...
0
votes
1
answer
34
views
Help decipher the notation said to denote a common pattern in various branches of science in Prelude to Mathematics by W. W. Sawyer
In Section 1.2 - Nature's Favorite Pattern? (excerpted below) of Prelude to Mathematics by W. W. Sawyer (1982), he said mathematicians used the notation $\nabla^2 V$ to denote a pattern that occurs &...
0
votes
2
answers
326
views
Is there any difference in superscript and subscript notation in finite difference method
Is there any difference in superscript and subscript notation in the finite difference method?
I see the same paper use (superscript for $x$ and superscript for $y$ notation) and (subscript for x and ...
2
votes
3
answers
109
views
Small doubt on derivatives acting on kets/bras
I have a quick, silly question. If $\psi(x):=\langle x|\psi\rangle$, does the bra $\langle x|$ 'go through' the $\partial_x$ operator, as in $$\langle x|\partial_x|\psi\rangle=\partial_x\psi(x) \quad ?...
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 ...
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'(...
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 ...
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 ...
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{[\...
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 ...
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 ...
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 ...
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
$$\partial_\mu x^\mu = -\partial_1x^1 + \partial_2x^2 + \partial_3x^3 + \partial_4x^4 \\
=-1+1+1+1\\
=...
3
votes
0
answers
546
views
Commutation relation of four vectors [closed]
I was trying to prove that: $$[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 ...
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:
$$\dot{q}=hA\Delta T\tag{1}$$
$$\dot{q}=\frac{\mathrm{d} Q}{\mathrm{d}t}\tag{2}$$
$$\dot{q}=\frac{\delta Q}{\mathrm{d}t}\tag{3}$$
$$\...
1
vote
2
answers
353
views
Are there differences in notation for the d'Alembert operator?
On Wikipedia the d'Alembert operator is defined as
$$\square = \partial ^\alpha \partial_\alpha = \frac{1}{c^2} \frac{\partial^2}{\partial t^2}-\nabla^2 $$
However, my professor uses the notation:
$$ \...
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.
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....
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 ...
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 ...
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 ...
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 ...
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)&...
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 ...
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:
$$
\text{Now we find from the commutation relation of } \partial_x \equiv \frac{...
2
votes
3
answers
193
views
Is $ d \mathbf v · d \mathbf v = d \mathit v^2 $?
My teacher has proved the following:
$$ \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} $$
Because ...
0
votes
1
answer
1k
views
What is $D$ or $D$-with-a-slash-through-it in the Standard Model equation(s)?
In the mathematical formulation of the Standard Model, which I do not understand yet, there is a capital letter $D$ or $D$-with-a-slash-through-it that I can't find an explanation for.
Flip Tanedo (a ...
-1
votes
2
answers
603
views
What does $d$ stand for in this formula?
Context: I am building a tennis ball machine and am having trouble interpreting the following formula for the flight path of the ball. I know all of the other values in the formula but the source I am ...
14
votes
3
answers
1k
views
What is meant by a partial derivative of a ket?
In my QM book I often see partial derivatives mixed with kets, like
$$
\frac{\partial}{\partial a} |\psi \rangle
$$
where $a \in \{x, y, z\}$. Here I'm assuming that $| \psi \rangle \in \mathbb{C}^n$ ...
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) ...
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 ...
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 ...
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 ...
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^{...
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 ...
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 ...