For the function f(x) given by:
f(x) = e^{2x} (x<0), = e^{-x} (x>0)
I have got the complex Fourier Transform to be:
F(k) = {3(k^{2} + ik + 2)}/{(k^{2}+1)(k^{2} + 4)}
Now I'm trying to verify the formula for the inverse transform by using a D-contour integral. Just taking the x>0 case I...
Hey! I am using c++ to simulate charges dissipating. I've written the output to a file in the form of a matrix, so in the file is just a matrix of numbers showing equipotentials etc.
I want to show the contours by using MATLAB. I'm not used to working in MATLAB but I tried! I clicked file...
I need to find all the separated solns of
x^2 \frac{\partial^2 u}{\partial x^2} + x\frac{\partial u}{\partial x} + \frac{\partial^2 u}{\partial y^2} = 0
in the strip {(x,y) : 0 < y < a, -\infty < x < \infty }
the separated solns must also satisfy u = 0 on both the edges, that is, on...
sorry it was meant to be..
\frac{(\lambda -1)a}{(x^2 + y^2 + z^2)^{3/2}} = \frac{(\lambda -1)a}{|\underline{r}|^3}
I forgot the 3/2 power which didn't make it a vector as you pointed out!
If this is wrong I can post my working (but its tedious to keep latexing my results!)
I get the expression down to \frac{(\lambda -1)a}{x^2+y^2+z^2}
so could I just say that if \lambda = 1 this would give " \underline{F}" say to be zero which implies dF/dx, dF/dy, dF/dz are all zero so...
Following on I'm trying to find the value of \lambda which makes
\frac{\lambda\underline{a}}{|\underline{r}|^3} - \frac{(\underline{a}.\underline{r})\underline{r}}{|\underline{r}|^5}
solenoidal. Where a is uniform.
I think I have to use div(PF) = PdivF + F.gradP (where P is a scalar...
Yeah I was being daft, after sleeping on it I came up with:
\underline{F} = [{(x^2 + y^2 + z^2)}^{n/2}x, {(x^2 + y^2 + z^2)}^{n/2}y, {(x^2 + y^2 + z^2)}^{n/2}z]
That's better, yeah?
so
\underline{F}_x = {(x^2 + y^2 + z^2)}^{n/2}x
\underline{F}_y = {(x^2 + y^2 + z^2)}^{n/2}y...
I want to find which values of n make the vector field
\underline{F} = {|\underline{r}|}^n\underline{r} solenoidal.
So I have to evaluate the divergence of this vector field I think, then show for which values of n it is zero?
Im starting by substituting:
\underline{r} = \sqrt{x^2...
Do I need to change the order of integration then and have new limits or can I choose to rearrange it to a more convenient form, like
\iiint {\sqrt(R^2 - 2aR\cos\theta + a^2)} R^2 sin\theta\,d\theta\,dR,d\phi
then integrating wrt \theta by parts?
\iiint {\sqrt(R^2 - 2aR\cos\theta + a^2)} R^2 \sin\theta\,dR\,d\theta\,d\phi
with the integration over R between 0 and a
the integration over between 0 and pi
the integration over between 0 and 2pi
Should I use integration by parts or should I take the R^2 sin(theta) under the square...
Yeah, I've got the distance AP to be
(R^2 - 2aR\cos\theta + a^2)^(1/2)
which I think is correct?
Now writing the triple integral of AP over the sphere R less than or equal to a in terms of spherical polar coords gives:
\iiint {\sqrt(R^2 - 2aR\cos\theta + a^2)} R^2...
Okay so I need:
(R\sin\theta\cos\phi)^2 + (R\sin\theta\sin\phi)^2 + (R\cos\theta)^2
How can I simplify this? In a book I have it just sets a "similar" expression equal to 1 with no intermediate steps:
(\sin\theta\cos\phi)^2 + (\sin\theta\sin\phi)^2 + (\cos\theta)^2 = 1
Also this...
Yeah, sorry forgot to mention that I had interchanged them.
So now I have two points in cartesian coordinates:
P (R\sin\theta\cos\theta , R\sin\theta\sin\phi , R\cos\theta)
A (0, 0, a)
In order to work out the distance AP do I need to square that awful looking thing?!