Awesome Solution to a MIT Multivar Problem
by Wolstenholme, Sep 3, 2015, 4:23 AM
OK, so in the last few days BOGTRO and I tested out of MIT's multivariable calculus class (class 18.02). One of the harder problems on the test was the following:
Consider the ball of radius
centered at the point 
. What is the average distance from the origin to a point in this ball?
So, the natural inclination would be to do a simple triple integral after translating to spherical coordinates. However, despite the nice bounds
and 
, the bounds for 
require the Law of Sines/Cosines which is just dumb. So, BOGTRO and I (with Chris Shao also aiding in the conception of this beautiful idea) present the following "clearly-intended" solution - prepare your bodies!
Unfortunately, BOGTRO and I kept up messing up our calculations so this took us an hour lol.
Consider the ball of radius
centered at the point 
. What is the average distance from the origin to a point in this ball?So, the natural inclination would be to do a simple triple integral after translating to spherical coordinates. However, despite the nice bounds
and 
, the bounds for 
require the Law of Sines/Cosines which is just dumb. So, BOGTRO and I (with Chris Shao also aiding in the conception of this beautiful idea) present the following "clearly-intended" solution - prepare your bodies!
(we could have easily used 
instead of 
is sent to a half-space whose boundary lies 
above the 
-
is sent to the point 
where![\[ u = \frac{r^2x}{x^2 + y^2 + z^2} \: \: v = \frac{r^2y}{x^2 + y^2 + z^2} \: \: w = \frac{r^2z}{x^2 + y^2 + z^2} \]](http://latex.artofproblemsolving.com/7/1/d/71d826c5c024b80f3986498a7c8587f703f3a50b.png)
![\[ \frac{d(u, v, w)}{d(x, y, z)} = \frac{r^6}{(x^2 + y^2 + z^2)^6}\begin{vmatrix}y^2 + z^2 - x^2 && -2xy && -2xz\\-2yx && z^2 + x^2 - y^2 && -2yz\\-2xz && -2yz && x^2 + y^2 - z^2\end{vmatrix} = -\frac{r^6}{(x^2 + y^2 + z^2)^3} \]](http://latex.artofproblemsolving.com/9/1/2/912accce53ee6d3d6a3ffac6905973f2fda584f0.png)
so 
and also 
.
Unfortunately, BOGTRO and I kept up messing up our calculations so this took us an hour lol.
August 2016
June 2016