Possibility of representing the Möbius strip by ways other than specifying a parameterization?

Question

I have looked up some texts, they all represent the Möbius strip by ways of specifying a parameterization. This is the only way how the Möbius strip is represented founded in the Wikipedia currently.

Question 1:

Can the Möbius strip be represented as the graph of a map?

For example, a map $\phi$ from a subset $K$ of $\mathbb{R}^2$ to $\mathbb{R}$ is specified, and the Möbius strip is exactly of the form $\{(V,\phi(V)) : V$ $\in$ $K \subseteq \mathbb{R}^2 \}$, which is called the graph of $\phi$.

Question 2:

Can the Möbius strip be represented as the locus defined by constraint equations? (With restrictions on allowable range of the variables.)

For example, an equation $f(X)=0$ is specified, where $X$ is in $\mathbb{R}^3$, $f$ map $\mathbb{R}^3$ to $\mathbb{R}$, and the Möbius strip is exactly of the form $\{X \in \mathbb{R}^3 : f(X)=0 \}$, which is called the locus defined by the constraint equation $f(X)=0$.

Answer 1

Since the Möbius strip does not live in $\mathbb{R}^2$ the first is not possible. To have it as a function $f(x,y,z)=0$see:

$$-R^2y+x^2y+y^3-2Rxz-2x^2z-2y^2z+yz^2=0.$$

I copied it from http://mathworld.wolfram.com/MoebiusStrip.html trula

Answer 2

Neither kind of representation is possible, because the Möbius strip is not orientable.

Suppose $M$ is a regular level set of a smooth function $F: U \to \mathbb R$, where $U$ is an open subset of $\mathbb R^3$. (A regular level set in this context is a level set on which the gradient of $F$ does not vanish.) Then $\operatorname{grad} f$ is a nowhere-vanishing normal vector field along $M$, so $M$ must be orientable.

Similarly, suppose $M$ is the graph of a smooth function $f: V\to \mathbb R$, where $V$ is an open subset of $\mathbb R^2$. Then $M$ is a regular level set of the function $F(x,y,z)= z-f(x,y)$, so again $M$ must be orientable.