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[Multivariable Calculus] Implicit Function Theorem

  1. Oct 13, 2016 #1
    I am having trouble doing this problem from my textbook... and have
    no idea how to doit.

    1. The problem statement, all variables and given/known data

    I am having trouble doing this problem from my textbook...

    Show that the equation x + y - z + cos(xyz) = 0 can be solved for z = g(x,y) near the origin. Find dg/dx and dg/dy

    (dg/dx and dg/dy are partial derivatives)
    2. Relevant equations
    Implicit function theorem.

    3. The attempt at a solution
    I tried computing dg/dx and dg/dy like it told me but
    I think that isn't what its asking..
     
  2. jcsd
  3. Oct 13, 2016 #2

    Mark44

    Staff: Mentor

    I think that the trick here is to recognize that if x and y are both close to 0, then xyz is also close to zero, so what can you say about cos(xyz)?
     
  4. Oct 13, 2016 #3

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    You are given an equation of the form ##f(x,y,z)=0## and an initial point ##(0,0,z_0)## that satisfies it (where I will let you figure out the value of ##z_0##). The implicit function theorem states that for certain conditions on the derivatives of ##f## in a neighborhood of ##(0,0,z_0)##, the equation is solvable for ##z## in terms of ##(x,y)##, near ##(0,0)##. Does your given ##f## satisfies those conditions? Does the theorem apply to your function?
     
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