Integration Methods Beyond 3 Dimensions

Date: 11/12/96 at 12:50:27
From: Rick Mendoza
Subject: Integration Methods Beyond 3 Dimensions

Are there any resources on the WWW which are relative to calculating 
areas of, for instance, a hypercube?  I have taken an advanced 
Calculus course at SDSU (San Diego State University), but this 
curiosity was apparently beyond the scope of any professor's personal 
interest.

Sincerely,
Rick Mendoza

Date: 11/13/96 at 12:42:15
From: Doctor Tom
Subject: Re: Integration Methods Beyond 3 Dimensions

Hi Rick,

I don't know about resources on the Web, but you might consider 
looking in some different advanced calculus texts.

The easiest way to think about it is to notice the pattern as the 
dimensions increase.  See what the problem looks like in 1, 2, and 3 
dimensions, and it's usually clear how to go to 4 (or more).

For example, to think about a 4-D hypercube, think about "cubes" in 1, 
2, and 3 dimensions first.  In 1-D, it's a line segment, and the 
"volume" is the length.  In 2-D, it's a square, and the "volume" is 
the area -- length times width.  In 3-D, it's a cube, and the volume 
is the volume -- length times width times height.  So in 4-D, the 
volume will be length times width times height times extent (I don't 
know what you want to call the 4th dimension, so I just made up the 
word "extent").

Or another way to think about it is by looking at a mathematical 
definition of "cubes" in 1 through 4 dimensions:

1-D:   a < x < b

2-D:   a < x < b and
       c < y < d

3-D:   a < x < b and
       c < y < d and
       e < z < f

4-D:   a < x < b and
       c < y < d and
       e < z < f and
       g < w < h      (I use w for the 4th coordinate)

The "volumes" of the examples above are:

1-D:  (b-a)
2-D:  (b-a)(d-c)
3-D:  (b-a)(d-c)(f-e)
4-D:  (b-a)(d-c)(f-e)(h-g)

You can do the same thing for other shapes.  For example, what
is the definition of 'sphere' in the first four dimensions?  Here are
some reasonable definitions:

1-D:  x^2 < r^2
2-D:  x^2 + y^2 < r^2
3-D:  x^2 + y^2 + z^2 < r^2
4-D:  x^2 + y^2 + z^2 + w^2 < r^2

To find the volumes of these spheres, just integrate over the 
appropriate ranges:

1-D:   int(-r, r, dx)
2-D:   int(-r, r, dx)int(-sqrt(1-x^2), sqrt(1-x^2), dy)
3-D:   int(-r, r, dx)int(-sqrt(1-x^2), sqrt(1-x^2), dy)
       int(-sqrt(1-x^2-y^2), sqrt(1-x^2-y^2), dz)
4-D:   int(-r, r, dx)int(-sqrt(1-x^2), sqrt(1-x^2), dy)
       int(-sqrt(1-x^2-y^2), sqrt(1-x^2-y^2), dz)
       int(-sqrt(1-x^2-y^2-z^2), sqrt(1-x^2-y^2-z^2), dw)

The stuff above is meant to indicate repeated integrals, and
the integrand in all cases is the constant 1.

I hope this helps some.

-Doctor Tom,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/