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Homework Help: Multivariable Calculus: Manifolds

  1. Aug 4, 2015 #1
    1. The problem statement, all variables and given/known data
    Let ##M## be the set of all points ##(x,y) \in \mathbb{R}^2## satisfying the equation

    ##xy^3 + \frac{x^4}{4} + \frac{y^4}{4} = 1 ##

    Prove that ##M## is a manifold. What is the dimension of ##M##?

    2. Relevant equations


    3. The attempt at a solution

    I think this question it started by saying the following:

    ##\phi=xy^3 + \frac{x^4}{4} + \frac{y^4}{4} - 1##

    Not overly sure how do this question so any help in the right direction would be appreciated. Anyway, I got the partial derivatives:

    ##\frac{{\partial}\phi}{{\partial}x}=y^3 + x^3##

    ##\frac{{\partial}\phi}{{\partial}y}=3xy^2 + y^3##

    After here I'm stuck, I can't find any clear way of answering this. Thanks in advance for any help.
     
  2. jcsd
  3. Aug 4, 2015 #2
    What is your definition of a manifold? Do you know the implicit function theorem?
     
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