Should i ask such questions in Homework Forum?! :confused:
It's not a homework at all... just hoping that someone have more experience with such things...
Differential equations and integral transforms
Hi!
I have some general questions on using integral transforms for solving differential equations.
Also, I know that Fourier and Laplace transforms are useful means for solving linear ODE's and PDE's.
1. Are there cases, when one of them is...
Hallo!
Homework Statement
Consider electrostatic Potential of point charge at point (0,0,0)
\phi = 1/r
I'm trying to calculate \Delta\phi
Homework Equations
The Attempt at a Solution
Actually it's not a difficult problem outside (0,0,0):
\nabla\phi = -\frac{\vec r}{r^3}
\Delta\phi = 0
But...
The capacitor of capacitance C is charged by battery (emf = E) (assume that there is no resistance). In the end charge of the capacitor Q = C*E; Work done by the battery W = Q*E=C*E^2. But The energy of charged capacitor is U = (C*E^2)/2. Work doesn't equals to stored energy. Where we lost...
So, i tried to get some more terms of \sin x expansion. Here is my result:
\sin x = 2 \sin \frac{x}{2} \cos \frac{x}{2} = 2 \sin \frac{x}{2} (\cos \frac{x}{2} -1 +1) = -2 \sin \frac{x}{2} (1-\cos \frac{x}{2}) + 2\sin \frac{x}{2} = -4 \sin \frac{x}{2} (\sin \frac{x}{4})^2 +2\sin \frac{x}{2}...
Possibly, Benorin meant \lim_{x\rightarrow 0} \frac{\sin(x)}{x}=1,
but i believe, that this can't help in solving initial problem. I have to use further terms of sinx series. \sin x = x - \frac{x^3}{3!}+... But how can i get them algebraically??
Hello!
\lim_{x\rightarrow 0} \frac{-x(1 -\cos x)}{\sin x - x}
I solved this limit using L'Hopital and expanding trigonometric functions to series.
But i have to solve it using algebraic tools (without series). I don't know how to do it. \sin x - x looks difficult to deal with.
I think you aren't right here. Distance to the axis doesn't remains the same for each point (it doesn't equals the distance to the centre of disc, but shortest distance to the axis!).
If you know I according to 3 orthogonal axis, then you can calculate I for each axis (through the same point)...
Are state functions defined for entire thermodynamics or for certain processes? If last then it's possible to say that in adiabatic process work is state function.