Multivariable Calculus: Manifolds

[teme92]

Homework Statement

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 $M$?

Homework Equations

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.

 

[micromass]

What is your definition of a manifold? Do you know the implicit function theorem?