Dec 09 2009
Where are your mathematics now?
Last week we pulled out the Mathematics Notes book from secondary school1. This was the book in which we had to write all the definitions and meanings for what we learned; and it had to be written in ink, for some reason that escapes me now.
Can you remember the formula for the volume of a sphere? Could you derive it if you had to? I’m happy to say I remembered it but I haven’t had a go at pulling it together from first principles. It’s all that integration, innit?
If you had kids in ten or twenty years’ time that were learning this stuff do you think you could help them with their homework?
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What?! We’d been drinking! Isn’t that what everyone does when the wine bottle’s been uncorked? ↩
3 Responses to “Where are your mathematics now?”
The infinitesimal volume element is dx dy dz. A sphere is defined by x^2 + y^2 + z^2 = R^2. So either you integrate dxdydz over the space defined by that constraint. e.g. x goes from 0 to sqrt(R^2 - y^2 - z^2), then similarly y going from 0 to sqrt(R^2 - z^2) and finally z from 0 to R. Or you do it in a spherical coordinate system. It’s left as an exercise for the interested reader to show that in spherical coords, the volume element is r^2 sin t dr dt dp.
Where r goes from 0->infinity, t from 0->pi, p from ->2pi.
Integrating this over a sphere of volume R is easy and gives:
R^3/3 2 * 2pi = 4/3 pi R^3 (as expected).
Doing it without calculus is harder.
Deriving the volume of a sphere in N dimensions is also possible. It turns out a sphere of a given radius has a maximum volume in a finite (non-integer) dimension. Which isn’t immediately obvious.
Lawrence (who didn’t have to look this up, because he still does maths on a regular basis!)
Oh, and spot the mistake in the cartesian example. I’ve only worked out the volume of an 8th of a sphere, so you need to multiply by 8 to get the volume of the whole sphere.
I’m not sure physicists and mathematicians are actually allowed to answer this one. Instead Lawrence must tell us some vital law or pertinent nugget from a subject he has not looked at in eight or ten years. But we’d have to know what he studied to ask him; obviously any information volunteered is not really a test of the question!