My textbook defines it like this:
Let F be a continuous function from a subset D(F) R2 into Rq. Suppose that D(F) is pathwise connected, and that every point in D(F) is either an interior point of D(F), or on the boundary of the interior of D(F). Then F is called a parametrized surface in Rq.
I understand what everything in the definition means and understand how to check for all the conditions. However, I do NOT understand what type of surface is being described. How do these special qualities of the domain enhance our understanding of the surface?
Can someone explain what a parametrized surface is (intuitively, not formally) and how that relates to this formal definition?
What consequences are their to a surface being a "parametrized surface?"
There are three ways I like to think about surfaces: 1) as graphs of functions from $\mathbb{R}^2$ to $\mathbb{R}$, 2) as level sets of functions $\mathbb{R}^3 \rightarrow \mathbb{R}$, and 3) as images of maps $\mathbb{R}^2 \rightarrow \mathbb{R}^3$. Surfaces of the third type are called parametric.
Here are examples of each:
1) The graph of $z = x^2 + y^2$ is a paraboloid.
2) The sphere is the level set $x^2 + y^2 + z^2 = 1$. Here the function from $\mathbb{R}^3$ to $\mathbb{R}$ is $F(x,y,z) = x^2 + y^2 + z^2$. Note that the graph of $F$ is a hypersurface in $\mathbb{R}^4$, but that when we take a level of $F$ (i.e., set it equal to a constant) it becomes a surface in $\mathbb{R}^3$.
3) The torus (surface of a donut) is a parametric surface that can be defined as the image of the map $G:\mathbb{R}^2 \rightarrow \mathbb{R}^3$ given be $G(u,v) = ((2 + cos(v))cos(u), (2 + cos(v))sin(u), sin(v))$. Here, the input variables $u$ and $v$ are called parameters, and the output coordinates could be specified as three "parametric equations":
x = (2 + cos(v))cos(u)
y = (2 + cos(v))sin(u)
z = sin(v)
As $u$ and $v$ take on different values, the function $G$ traces out the surface in space.
There's more to say here, but that's a start!