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Vector Calculus (Surfaces)

  1. May 14, 2014 #1
    1. The problem statement, all variables and given/known data

    (a) Show that the four points r1 = (1, 0, 1), r2 = (4, 3, 5), r3 = (6, 4, 6) and
    r4 = (3, 1, 2) are coplanar and the vertices of a parallelogram. Let S
    be the closed planar region given by the interior and boundary of this
    parallelogram. An arbitrary point of S can be written as the convex linear
    combination

    [itex]\sum a_{j}r_{j}[/itex] for j= 1 to j=4 [itex]\sum a_{j}=1[/itex] [itex]0<a_{j}<1[/itex]

    Show that the vertices, edges and interior of the rectangle R = [0, 1]×[0, 1]
    are mapped onto the vertices, edges and interior of S by the linear map
    (parametrization) r = r(u, v) : R → S

    r = (x, y, z) = (1 + 3u + 2v, 3u + v, 1 + 4u + v), (u, v) ∈ [0, 1] × [0, 1]

    2. Relevant equations

    Not sure

    3. The attempt at a solution

    I've shown that the four points are coplanar and the vertices of a parallelogram, however I really have no idea about the rest. Some guidance would be very much appreciated!
     

    Attached Files:

    Last edited: May 14, 2014
  2. jcsd
  3. May 14, 2014 #2
    Sorry! I just saw that we cant use attachments to present question. I've edited the above post with the full question.
     
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