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Implicit Mapping Theorem for a Surface

  1. Apr 7, 2013 #1
    The question, from Edward's Advanced Calculus (which is becoming a rather frustrating book), asks if the following surface can be represented as a function of ##z(x,y)## near the point ##(0,2,1)##:

    [tex]xy - ylog(z) + sin(xz) = 0[/tex]
    Naturally, this should invite me to use the Implicit Mapping Theorem, but I've seen this theorem presented mostly for systems of equations, rather than a single case. But anyhow, let the surface be called ##G({\bf{x}},{\bf{y}})##, where ##{\bf{x}} = (x,y)## and ##{\bf{y}} = z##. Then,
    [tex]\frac{\partial{G}}{\partial{z}} = xcos(xz) - \frac{y}{z}[/tex]
    So at ##(0,2,1)##, ##G'((0,2),(1)) = -2##. Thus, since the invertible "matrix" is non-singular, there exists a neighborhood near ##(0,2,1)## for which the surface can be represented as a function of ##z##.

    I suspect this is correct, but being confused by some notation, I just wanted to make sure. Thanks.
     
  2. jcsd
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