All Questions
8 questions
1
vote
3
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176
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Where to apply $\nabla$ operator when taking curl of a cross product?
In my EM class we went over $$\nabla\times \frac{\vec{d}\times \vec{r}}{r^3}$$ which apparently can be breaken down to $$r(d\cdot \nabla)\frac{1}{r^3}-d(r\cdot\nabla)\frac{1}{r^3}+\frac{\nabla\times(d\...
- 33
1
vote
1
answer
170
views
What does it mean to differentiate a scalar with respect to a vector?
I am reading the special relativity lecture notes that I got from a professor of mine. It says that the Lagrangian is
$$L = \frac{1}{2}m|\dot{\boldsymbol{x}}|^2 - V(\boldsymbol{x}) \tag{1}$$
The notes ...
- 1,254
2
votes
1
answer
103
views
What is the meaning of the del operator in this equation?
$$\frac{\partial \left(\rho_m \vec{v}_m \right)}{\partial t} + \nabla \cdot \left(\rho_m \vec{v}_m\vec{v}_m \right) \\ = - \nabla P_m + \nabla \left(\mu_m \nabla \vec{v}_m \right) + \nabla \left(\...
- 55
-1
votes
3
answers
180
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Avoiding a confusion with dot product
Some days ago I have asked a question about a formula for power, many generous people have answered my question and clarify for me that the correct formula of work is
$$\mathrm{d}W= \mathbf{F}\cdot \...
4
votes
1
answer
167
views
What motivates defining vectors as first order differential operators?
I have read some introductions to geometrical ideas and tensors and physics and what some of them do (see, for example, Frankel's Geometry of Physics) is define a vector as a first order differential ...
- 73
13
votes
7
answers
3k
views
Can we divide a vector by another vector? How about this: $a = vdv/dx?$
My physics teacher told us that we can’t divide vectors, that vector division has no physical meaning or significance. How about this: $$a = vdv/dx.$$
It says acceleration vector equals velocity (as ...
- 876
15
votes
5
answers
2k
views
What does it mean for a physical quantity if its mixed second partial derivatives are not equal?
This goes for every problem (either in electromagnetism or fluid dynamics) that has to do with vector fields. Say we have a fluid flowing in a closed circular pipe (or an electromagnetic field, the ...
- 7,860
7
votes
6
answers
8k
views
How is gradient the maximum rate of change of a function?
Recently I read a book which described about gradient. It says
$${\rm d}T~=~ \nabla T \cdot {\rm d}{\bf r},$$
and suddenly they concluded that $\nabla T$ is the maximum rate of change of $f(T)$ ...
- 872