Homework 7, Due by email to TA Nov 7 Solution

1. (1 pt) What is the normal vector to the sphere {$x^2+y^2+z^2=1$} at the

  surface point {$(x,y,z)$}?

The normal vector at the surface is the gradient vector grad({$x^2+y^2+z^2=1$})= (2x,2y,2z) The sign only indicates if it goes inside or goes outside the surface so both are acceptable.

2. (1 pt) What is the normal vector to the paraboloid {$z=x^2+y^2$} at the

  surface point {$(x,y,x^2+y^2)$}?

The normal vector at the surface is the gradient vector grad({$z=x^2+y^2$})=(2x,2y,-1)

3. (4pt, this question will take some thinking).

  Imagine that you have an infinitely large room illuminated
  by one infinitely long row of point lights.  This figure
  shows a side view of the room.

  The lights are h above the floor and are 1
  meter from each other.  Assume that the ceiling above the
  lights is black and that no light reflects off of anything.

  An object at distance d from a light gets illuminated
  with a brightness 1/d² .

  Each point on the floor is illuminated by all the lights,
  but more brightly by the closer lights.  

  A point p directly below a light will be a little
  brighter than a point q halfway between two such
  points.  That is the problem --- we want the floor (at least
  the part directly below the line of lights) to be evenly
  lit, at least within 20%.

  However, the higher the line of lights, the more evenly the
  floor will be lit.

  Your question is to tell us what is the minimum value for
  h so that the line of the floor below the line of
  lights is evenly lit within 20%.

  E.g., the brightness at p is {$$ \sum_{d=-\infty}^{\infty} \frac{1}{h^2+d^2} $$}

h=0.49 for the point directly under a light to be 20% brighter. If you computed for the point in the middle being 20% dimmer, which is also ok, you'll get a slightly different answer.

There are many ways to do this. hw7q3.mw is a Maple worksheet, which you can play with. hw7q3.pdf is a printout you can read w/o running Maple. What I did:

b1 is the brightness of one light in the row, where you are directly under one of the lights.
b2 is the brightness of one light in the row, where you are between two lights.
s1 is the total brightness of the row, where you are directly under one of the lights. It is unusual for an infinite sum to have a closed form solution. Usually you have to use a numerical approximation. I constructed this problem to work out.
s2 is the total brightness of the row, where you are between two lights.
To get a sense of what's happening, I plotted s1 and s2.
Then I used solve. For this physical problem, h is a positive real number, but solve doesn't know that. So it also finds other extraneous solutions that are not physically valid. You often cannot blindly use general things like solve w/o using your also extra knowledge of the problem.
To double check, I plotted s1/s2.