Show that the set $M$ is a 2-dimensional $C^1$-manifold.

Question

Show that the set $M:=${$(x,y,z)\in \mathbb{R}^3|x^3+y^3+z^3=1$}$\subset\mathbb{R}^3$ is a 2-dimensional $C^1$-manifold. I don't really understand the definition of a manifold so I have no idea what I'd have to show. Thanks for any help.

Answer 1

I am looking at the 1973 edition of the text and it seems that the actual definition on a manifold is hidden away on page 110 in the second paragraph from the bottom. This is in chapter 2 section 5. I found the definition a little cryptic, so I sympathize with your confusion and will say a couple of words about it. Edwards defines a manifold as something that is locally the graph of a differentiable function. So, if you are working with this definition here is an outline of what a proof should look like. Basically, for each point $x\in M\subset \mathbb{R}^n$, you need to find a subset of $P$ of $M$ containing $x$, and an open subset $U$ of $\mathbb{R}^k$ and a differentiable function $h: U\rightarrow \mathbb{R}^{n-k}$, such that $P$ is the graph of $h$. Note that the graph can be identified with a subset of $\mathbb{R}^n$. However, it might not happen that $P$ is not literally the graph of the function, for example consider a vertical line in $\mathbb{R}^2$. This is why the definition allows you to permute the coordinates. In order to do the problem you have given, I recommend that you closely study the definition and make sure you understand all the terms in it. Then examine the worked examples to see what a proof looks like when written down.