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Multivaraible Calculus. surfaces in R2 andR3

  1. Oct 13, 2008 #1
    1. The problem statement, all variables and given/known data
    In class, we studied functions ⃗r : I → R^3, where I ⊂ R is some interval.
    Let us now consider a function
    ⃗r : U → R^3, U ⊂ R^2
    That is, we have a function, ⃗r, which sends a point (u, v) in the plane to a
    point (vector) in R3 . You may call it a “vector function of two arguments”
    or a “vector function of a vector argument”. To be specific:
    Let
    U = {(u, v), v > 0} ⊂ R^2
    ⃗r(u, v) = (√v/2 cosh u, √v sinh u, v)
    or, equivalently,
    x = √v/2 cosh u
    y = √v sinh u
    z = v

    1.Verify that the points ⃗r(u, v) satisfy the equation z = 4x^2− y^2
    .
    2.Identify the surface S given by the equation z = 4x^2-y^2
    Is it an ellipsoid, paraboloid, hyperboloid (and which one), cone, cyllinder?



    2. Relevant equations



    3. The attempt at a solution
    I know that the surface given by the equation is a hyperbolic parabloid, but I have no idea how to approach showing that the points satify the equation. When I tried just plugging the values of x,y, and z into the equation I end up with something very messy, and I'm not quite sure what I am supposed to be solving for. If anyone could explain this to me it would be great!!
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
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