All Questions
26 questions
-2
votes
2
answers
62
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Can the different differentiation notations be equated and do they have an integral definition? [closed]
Are these all equivalent and is there an extension of this to other notation?
Does anyone have a clear and concise chart equating the different notation dialects?
I am also curious if there are more ...
2
votes
1
answer
355
views
$\nabla$, $\cdot \nabla$, $\nabla \cdot$, $\nabla^2$ - What do they do? [closed]
I'm trying to teach myself Smoothed Particle Hydrodynamics. Unfortunately, my background is in electronics, so the Navier Stokes equations are somewhat alien to me, as is vector calculus. The video I'...
- 114
0
votes
1
answer
57
views
What does this vertical line notation mean?
Here is the definition of the Noether momentum in my script.
$$I = \left.\frac{\partial L}{\partial \dot{x}} \frac{d x}{d \alpha} \right|_{\alpha=0} = \frac{\partial L}{\partial \dot{x}} = m \dot{x} = ...
- 85
1
vote
1
answer
288
views
Question on how to make product rule for differentiation consistent with operators? [duplicate]
By the product rule for differentiation:$$\frac{\partial(\hat A\psi)}{\partial x}=\left(\frac{\partial\hat A}{\partial x}\right)\psi+\hat A\left(\frac{\partial\psi}{\partial x}\right)\tag{1}$$
Where $\...
2
votes
1
answer
89
views
Does the expression "$𝑑𝑠^2$..." mean the same thing as "$\Delta 𝑠^2$... "?
I reviewed this question but sometimes I'm unsure about delta ($\Delta$) versus differential ($d$) notation.
Does the expression "$ds^2=-c^2dt^2+a^2(t)[dr^2 + S_k^2(r)d\Omega^2 ]$" mean the ...
- 181
0
votes
0
answers
80
views
Why is cancellation of differnetial not allowed here?
This is about cancelation of differentials .I am learning basics of tesnor from "Mathematical Methods " by Boas. There I encountered this epression which author says are equal. $$ \frac{\...
- 128
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 \...
- 8,040
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 &...
- 109
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 ...
- 3
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'(...
- 373
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 ...
- 8,040
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....
- 31
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 ...
- 4,855
-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 ...
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 ...
- 15
0
votes
3
answers
624
views
Showing the equivalence between the chain rule's Leibniz and Lagrange Notations
This may seem more math related but this question crossed my mind as I was reading the derivation of the Euler-Lagrange Equation.
In math, we were introduced to the Lagrange notation of the derivative ...
- 97
-2
votes
3
answers
104
views
Changing derivative to difference quotient
Can differential be changed to Delta or difference? In high school education, in the acceleration section of Newton's formula 2, acceleration is a change velocity (velocity difference) divided by a ...
- 19
1
vote
0
answers
285
views
Why do they specify the differential of Work using an lowercase delta $\delta W$, instead of $dW$ [duplicate]
I was curious, why do they specify the differential of Work using an lowercase delta symbol $\delta$ as in "$\delta W$", instead of using a $d$, as in $dW$. For example:
$$\delta W=\vec{F} ...
- 111
2
votes
1
answer
253
views
Abuse of Calculus [duplicate]
I'm following Professor R. Shankar's Fundamentals of Physics course on YouTube.
There I saw him doing manipulations of Calculus I never saw before.
Here it goes,
$$\newcommand\deriv[2]{\frac{\mathrm ...
- 21
2
votes
1
answer
3k
views
Trying to understand the difference between $\Delta t$ and $dt$ [duplicate]
I'm trying to gain a more conceptual understanding of derivatives and would appreciate your feedback on this.
Say I have a quantity, $x$, at time $t$. Now $x$ moves to a different location $x'$ in ...
- 197
-1
votes
1
answer
651
views
What is difference between $d\vec{l}$ and $\vec{dl}$? [closed]
What is difference between $d\vec{l}$ and $\vec{dl}$? $d$ means differential as usual.
- 15
6
votes
2
answers
5k
views
What does $(\mathbf{u}\cdot\nabla)\mathbf{u}$ mean in the Navier-Stokes equation?
I am studying the Navier-Stokes equations and I have the equation in the form:
$$\rho \dfrac{\partial{\mathbf{u}}}{\partial{t}} + \rho (\mathbf{u}\cdot\nabla)\mathbf{u} - \mu\nabla^2\mathbf{u} + \...
0
votes
3
answers
577
views
What is meant by $dy/y$?
Consider the language in the following example:
What is meant by $dg$ and $dR$, and also by $dg/g$? Why does $dR/R=-2/100$ (negative for shrinks)? Is $4\%$ unity change? I mean $dg/g=4\%$ or $dg=...
- 93
2
votes
1
answer
5k
views
Two different meanings of $\nabla$ with subscript?
I am trying to understand the meaning of $\nabla$ when it appears with subscript. I have found two separate Physics SE answers that imply different meanings.
The notation $\vec \nabla_B$ means ...
0
votes
1
answer
2k
views
Use of infinitesimals in physics [duplicate]
I want to ask about infinitesimals and non-standard analysis. In physics we always use $\mathrm dx,~\mathrm dv,~\mathrm dt$ etc. as infinitesimal quantities. When we deduce equations in physics, when ...
- 41
3
votes
2
answers
5k
views
Feynman's subscript notation
Consider this vector calculus identity:
$$
\mathbf{A} \times \left( \nabla \times \mathbf{B} \right) = \nabla_\mathbf{B} \left( \mathbf{A \cdot B} \right) - \left( \mathbf{A} \cdot \nabla \right) \...
- 163