All Questions
13 questions
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0
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40
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Mass Conservation in Kinetic Theory
In chapter 9 (The Boltzmann Equation) of Schwabl's 2006 text 'Statistical Mechanics', the author has the following statement of conservation of mass,
$$ \frac{\partial n}{\partial t} + \nabla \mathrm{...
2
votes
1
answer
2k
views
Meaning of the transpose of a gradient
Sometimes I encounter PDE's with a term like this
$\nabla \cdot c(\nabla \textbf{v} + (\nabla \textbf{v})^T)$
An example are the Navier-Stokes equations. Oftentimes this can be further simplified to $...
- 25
7
votes
3
answers
1k
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Vector cross product formula without a second term (Spiegel, Theoretical Mechanics)
In Spiegel's Outline Of Theoretical Mechanics (more precisely in the Moving Coordinate Systems chapter, § "Derivative Operators") I find (both in the 1968 and the 1977 edition) the following ...
2
votes
3
answers
193
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Is $ d \mathbf v · d \mathbf v = d \mathit v^2 $?
My teacher has proved the following:
$$ \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} $$
Because ...
- 23
0
votes
1
answer
197
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4-velocity lowering index question
The 4-velocity in contravariant form is given by
$$V^\mu=\frac{dx^\mu}{d\tau}$$
for some general co-ordinates $x^\mu$ and proper time $\tau$.
Is the 4-velocity in covariant form given by
$$V_\nu=V^\...
- 5,991
1
vote
1
answer
415
views
Vector calculus notation, maybe?
I just got a new book on turbomachinery that uses some notation I'm not familiar with.
$$ \nabla \lor \vec{W} = -2\vec{\Omega} $$
The del-(something)-vector format makes me think its vector calculus....
- 187
1
vote
1
answer
2k
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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 $-...
-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
3
votes
2
answers
134
views
Generalization of $F=mv\frac{dv}{dx}=\frac{m}{2}\frac{d}{dx}(v^2)$ to 3-dimensions in a compact notation
Starting from $F=ma=m\frac{dv}{dt}$, in 1-dimension, it is easy to show that $$F=mv\frac{dv}{dx}=\frac{m}{2}\frac{d}{dx}(v^2).\tag{1}$$ Can we generalize this formula in 3-dimensions? In 3D, $$\textbf{...
- 12.3k
7
votes
4
answers
16k
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Is Del (or Nabla) an operator or a vector?
Is Del (or Nabla, $\nabla$) an operator or a vector ?
\begin{equation*}
\nabla\equiv\frac{\partial}{\partial x}\vec{i}+\frac{\partial}{\partial y}\vec{j}+\frac{\partial}{\partial z}\vec{k}
\end{...
- 129
2
votes
3
answers
179
views
Difference between $|d{\bf r}|$ and $d|{\bf r}|$
What is the difference between $|d{\bf r}|$ and $d|{\bf r}|$ and why are both of them not always equal to each other?
My question might seem stupid to some and will probably get downvoted but I have ...
- 733
1
vote
2
answers
4k
views
Is there a difference in handwritten nabla $\vec{\nabla}$ with an overset arrow and typeset nabla $\nabla$?
According to some physicist at KIT it is usual to write the following when using pen and paper:
whereas in typeset texts you write $\nabla$.
Is that true? Are there sources for this convention?
- 955
1
vote
4
answers
9k
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Dot product of vector and its derivative with respect to time? How does $L \cdot\frac{dL}{dt} = \frac{1}{2}\frac{d(L^2)}{dt}$? [closed]
How does:
$$L \cdot\frac{dL}{dt} = \frac{1}{2}\frac{d(L^2)}{dt}$$
where L is a vector (I dunno how to make it bold in the equation).
How do they reach to this right hand side equation?
And what is ...
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