Homework 7, Due by email to TA Nov 6

1. (1 pt) What is the normal vector to the sphere {$x^2+y^2+z^2=1$} at the surface point {$(x,y,z)$}?
2. (1 pt) What is the normal vector to the paraboloid {$z=x^2+y^2$} at the surface point {$(x,y,x^2+y^2)$}?
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 {$$ \sum_{d=-\infty}^{\infty} \frac{1}{h^2+d^2} $$}