# Multivariable Calculus: Manifolds

## 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##?

## 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.

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