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Finding corrections to volume of a two-sphere in 3-sphere geometry

  • Thread starter Raziel2701
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Homework Statement


Given the volume of a 2-sphere in a 3-sphere geometry,
[tex]V=4\pi a^3[\frac{1}{2}\arcsin (r/a) -\frac{r}{2a}[1-(\frac{r}{a})^2]^{(1/2)}][/tex], derive the 1st order correction when r/a<<1

I am also given that this is approximately [tex]V\approx\frac{4\pi r^3}{3}[1+()][/tex]

where inside the blank parenthesis should read "corrections of order (r/a)^2


The Attempt at a Solution


So since r/a<<1 I figure that the only important term should be the arcsine and that I should expand it, but if I neglect everything but the arcsine I don't get the approximate expression, I just get the 4pir^3 times the first few terms of the arcsine expansion, not to mention that I am not entirely sure what is meant by corrections of order (r/a)^2 since the arcsine expansion has odd powers.
 

Answers and Replies

  • #2
tiny-tim
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Hi Raziel2701! :smile:

(have a pi: π and a square-root: √ and try using the X2 icon just above the Reply box :wink:)
So since r/a<<1 I figure that the only important term should be the arcsine and that I should expand it, but if I neglect everything but the arcsine I don't get the approximate expression, I just get the 4pir^3 times the first few terms of the arcsine expansion, not to mention that I am not entirely sure what is meant by corrections of order (r/a)^2 since the arcsine expansion has odd powers.
But arcsine(r/a) is approximately r/a, so it's almost the same size as the other part :confused:

you need to expand both of them :redface:
 

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