This page
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/capeng2.html
talks about how the battery does work to move a charge from one plate to the other plate. Can charge jump across the gap between the plates? I was under the impression that the plates get charged due to electrons...
Hi,
Does the Descartes rule of signs count multiplicities when giving its upper bound for roots? That is if I have 3 sign changes, does that mean there is a maximum of 3 positive roots counting multiplicities or not counting multiplicities?
Thanks
differentiation of a functional
Where \phi = \phi(x) and the functional F=F(\phi(x)) = \int d^d x [\frac{1}{2}K^2(\bigtriangledown\phi)^2+ V (\phi)]
, the author says the derivative with respect to phi gives
\frac {\partial F} {\partial \phi(x)} = -K^2\bigtriangledown^2\phi + V'(\phi)...
Hi,
If I have the equation
y' = ax - by
where y = y(t) , x= x(t)
and y' = \frac{dy}{dt}
then what is
\frac {d}{dy} y' = \frac {d}{dy}(ax - by)
?
I think it would come out to
\frac {dy'}{dy} = a \frac {dx}{dt}\frac {dt}{dy} - b
Is that right? In general...
I think it has to do with the laplacian being rotationally invariant - hence no theta dependence. See sec 8.3/8.4:
http://www.math.ucsb.edu/~grigoryan/124B/lecs/lec8.pdf
could anyone else explain this a bit better? Is that true - if a PDE is rotationally invariant, its solutions will have no...
Hi,
When can I assume that the solution the laplace equation (or poisson equation) is radial? That is, when can I look only at (in polar coordinates)
\frac{\partial u^2} {\partial^2 r} + \frac{1}{r} \frac{\partial u} {\partial r} = f(r,\theta)
instead of
\frac{\partial u^2} {\partial^2 r}...
Hi,
So if I start with the boundary conditions
U(0,t) = T1 and U(L,t) = T2
and T1 does not equal T2, it seems that you are supposed to look at the 'steady state solution' (solution as t goes to infinity)?
which satisfies
T''(x) = 0
so the solutions are
T(x) = Ax + B
and then you...
Oh! ok that makes sense. But then what exactly do you mean by 'differential notation'? I see that if you 'multiply' by dx then that's what you get, but I know that that isn't really very rigorous and not completely correct since a differential isn't a fraction. Does the "d" mean in general a...
Actually now I'm looking back at this and I'm not sure I follow exactly what you mean. What are you taking the derivative of? Each side independently and then summing them? As in...
\frac {\partial} {\partial p} (pv)
and then
\frac {\partial} {\partial v} (pv)
etc...and then...
I'm looking at one step in my thermodynamics book and they go from
pV = \nu*R*T
to
p*dV + V*dp = \nu * R * dT
I think there's an application of the chain rule in here but I don't see exactly how it's working. Could someone show me the steps in between?
Thanks!
Hi,
I thought that if you integrate with limits, you don't include a constant, but if you don't integrate with limits (indefinite), there is a constant. But my book gives the example (all functions are single variable functions, initially of x but then changed to s for the integration)...