MATH 550
Professor Girardi
Quiz 9
Due the week of 23 April 2000

  1. Let
    F(x , y , z ) = < x , 0 , z > .
    Let S be the surface of the cube bounded by the 6 planes:
    x= 0 , y=0 , z=0 , x=1 , y=1 , z=1 ,
    and oriented by the outward normals.
    1. Find the flux of F across S , without using Gauss's Theorem.
    2. Find the flux of F across S , using Gauss's Theorem.
  2. Let
    F(x , y , z ) = < x2 , 0 , 2z > .
    Let:
    1. S1 be the surface of the hemisphere
      x2 + y2 +z2 = 9
      with z>= 0, oriented by upward pointing normals
    2. S2 be the surface
      { (x,y,0) : x2 + y2 <= 9 },
      oriented by downward pointing normal,
    3. S3 be the surface that is the union of S1 and S2 , orientation as before, thus outward pointing normals.
    1. Find the flux of F across S1 , without using Gauss's Theorem.
    2. Find the flux of F across S2 , without using Gauss's Theorem.
    3. Find the flux of F across S3 , using Gauss's Theorem.
      (Hint: the flux of F across S3 = flux of F across S1 +flux of F across S2 , thus you know the answer a prior).
  3. Problem 15 from section 8.1 (pg 477).
    (Hint: The book's Theorem 2 is in I.S. 8.a.1).
  4. Using Green's Theorem, find the line integral of F over c where:
    F(x , y ) = < 2x+3y+2 , -x + 4y - 3 >
    and c traces out the ellipse
    x2 + 4 y2 = 4 ,
    oriented clockwise.
    (Hint: problem 15 from section 8.1 comes in handy.)
    (Warning: be sure the negative signs in the definition of F printed off.)

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