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Questions for Assignment 3 - Honors Calculus III | MATH 2511, Assignments of Advanced Calculus

Seton Hall University (SHU)Advanced Calculus

Material Type: Assignment; Class: Honors Calculus III; Subject: Mathematics; University: Seton Hall University; Term: Unknown 1989;

Typology: Assignments

Pre 2010

Uploaded on 08/08/2009

koofers-user-syg
koofers-user-syg 🇺🇸

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Maple #3 – Optional
We want to investigate in which dimension the unit ball has the largest “volume”. Recall:
1-D “ball”: (which would be the interval )
2-D “ball”: (also known as the disk of radius r)
3-D “ball”: (a standard 3D ball of radius r)r
4-D “ball”:
…
n-D “ball”:
Drawing pictures of the 2D and 3D cases to determine the bounds, we know that:
“volume” of :
(length of the interval (-r,r))
Maple: >
“volume” of :
(area of disk radius r)
Maple: >
Volume of :
pf2

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Maple #3 – Optional

We want to investigate in which dimension the unit ball has the largest “volume”. Recall:  1-D “ball”: (which would be the interval )  2-D “ball”: (also known as the disk of radius r )  3-D “ball”: (a standard 3D ball of radius r )r  4-D “ball”:  …  n-D “ball”: Drawing pictures of the 2D and 3D cases to determine the bounds, we know that:

 “volume” of :

 (length of the interval (-r,r))

 Maple: >

 “volume” of :

 (area of disk radius r)

 Maple: >

 Volume of :

 Maple: >

I have used Maple to evaluate these integrals for us. Of course we could have also used polar coordinates, but Maple gives us the answers just fine. Moreover, there is a simple pattern here, so it should be easy to verify, using Maple, that, for example:

 Volume of

Your Assignment:

  1. Compute the volumes of the 1D, 2D, 3D, …, 10D balls of radius r
  2. Substitute r = 1 in the volumes found in step 1 and get a decimal answer for each n-ball
  3. Which of these unit balls has the largest volume? This would be good enough for this assignment. It is due Dec 17th, either via email or in person at 12:45pm. Technically, though, this is no proof since we only consider the first 10 dimensions. It turns out, however, that the “true” maximum volume of the unit n-ball occurs at a fractional dimension (whatever that is), the closest integer of which should be your answer. I’ll give you references for your enjoyment after the assignment is due.

For your Entertainment only:

The following is for your Christmas-break entertainment. You do not have to turn this in as part of the optional assignment. Let’s look at “volume” and “area” of 1D, 2D, and 3D balls:

  1. There seems to be an interesting relationship of the “volume” to the “surface area” of 2D, 3D, and 4D balls. What is it (consider r as the variable)?
  2. Compute the “surface area” of the remaining balls for which you know the volume, using Maple to do the computations.
  3. Does the relation you conjectured remain true in all of your examples?
  4. Prove (or find a counterexample for) your conjecture (not so easy). “Volume” “Area” of boundary 1D ball (interval) (^) (length of interval) (2 points) 2D ball (unit disk) (^) (standard area) (length of circle) 3D ball (see above) (surface area of sphere) 4D ball (see above) (verify)