Ah thanks! I've got there now. On the solution sheet they seem to suggest a different method using f''(r) but don't show it explicitly but this definitely seems valid. (Though I think the + should be a - in your third equation after splitting the fraction.)
Homework Statement
The final part of the problem I am trying to solve requires the proof of the following equation:
\frac{d}{dr}(\frac{rf'(r)-f(r)+f^2(r)}{r^2 f^2(r)})=0[/B]Homework Equations
I've been given the ansatz:
f(r)=(1-kr^2)^{-1}
leading to
f'(r)=2krf^2(r)...
So starting from 6.24, by bringing the b inside the square root, I can replace (1-b^2V(r))^{1/2} With (1-\frac{r_1^2}{r^2}(1-\frac{2M}{r}+\frac{2M}{r_1}))^{1/2}. Ignoring higher orders of M. Expanding this again however doesn't seem to get me near the right answer.
Hi, thanks for you reply
Sorry I made a mistake in my first statement, where I put my integral in section 3 tit should show t=\int_{r_1}^r(1+\frac{2M}{r}+\frac{b^2}{2}[\frac{1}{r^2}-\frac{4M^2}{r^4}])dr
Referring back to your solution though, is the argument not already in first order of M...
Homework Statement
The step I am trying to follow is detailed here where I am trying to get from equation 6.26:
t=\int_{r_1}^{r}(1+\frac{2M}{r}+\frac{b^2V(r)}{2}+\frac{Mb^2V(r)}{r})dr
to equation 6.30
t=\sqrt{r^2-r_1^2}+2Mln(\frac{r+\sqrt{r^2-r_1^2}}{r_1})+M(\frac{r-r_1}{r+r_1})^{1/2}
Homework...
Homework Statement
We have the equation
## (\frac{dr}{ds})^2+(\frac{l}{r})^2=1 ##
and want to solve to get ## r=\sqrt{l^2+(s-s_0)^2}##
Homework EquationsThe Attempt at a Solution
I have worked backwards, plugging in the solution to prove that it is correct, but the closest I have gotten to...
Homework Statement
In order to use cauchy's residue theorem for a question, I need to put
##f(x)=\frac{z^{1/2}}{1+\sqrt{2}z+z^2}##
Into the form
##f(x)=\frac{\phi(z)}{(z-z_0)^m}##.
Where I can have multiple forms of
##{(z-z_0)^m}##
on the denominator, e.g...
Yeah I think I understand that now, thanks a lot for your help, I see I need to be a lot more careful with what is just a component and what is a matrix. Hopefuly that will come with more practice of this notation.
Ok I see how that works now I think. As there's no implicit sum its just the component of a vector. Am I right in thinking though that a subscript usually indicates a row vector, and a superscript a column?
Homework Statement
I'm having trouble with understanding four vectors in particle physics.
I'm reading this wikipedia page,http://en.wikipedia.org/wiki/Einstein_notation, and its telling me that
## v^\mu= \begin{pmatrix} \mu_0 \\ \mu_1 \\ \mu_2 \\ \mu_3 \end{pmatrix} ##
and
## v_\mu=...
Homework Statement
(This isn't coursework, just a revision question)
Exercise: An ellipse can be defined as the locus of all points, P, in the plane such that
PF_1+PF_2=\frac{2p}{(1-ε^2)}
where F1 and F2 are two fixed points, and PF1 is the distance from P to F1 (similarly, P F2).
F1 and F2 are...