Question about parametrization and number of free variables

In summary, when a parametric equation has two free variables, it creates a surface, while a parametric equation with one free variable draws a line. This corresponds to the dimension of the geometric set, with one dimension representing a line and two dimensions representing a plane. Additionally, curves are considered 1-dimensional manifolds while surfaces are 2-dimensional manifolds.
  • #1
Hey guys, how come when you have a parametric equation with two free variables it creates a surface, but when you have a parametric equation with one free variable it draws out a line? I sort of get it intuitively, one dimension is just a line, two dimensions is a plane, so I guess this sort of corresponds to this effect, but is there a more rigorous explanation? I didn't know which section to put this in, sorry if it's not the right one.
 
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  • #2
coolbeans777 said:
Hey guys, how come when you have a parametric equation with two free variables it creates a surface, but when you have a parametric equation with one free variable it draws out a line? I sort of get it intuitively, one dimension is just a line, two dimensions is a plane, so I guess this sort of corresponds to this effect, but is there a more rigorous explanation? I didn't know which section to put this in, sorry if it's not the right one.
That's pretty much it. A geometric set has "dimension n" means that we can determine specific points in the set using n free variables or parameters.
 
  • #3
In a less natural/immediate way than HallsofIvy's comment, is the fact that curves

(reasonably-nice ones , at least ), are 1-dimensional manifolds, and surfaces are

2-D manifolds.
 

1. What is parametrization in science?

Parametrization in science refers to the process of representing a system or phenomenon using a set of parameters or variables. These parameters are used to describe the characteristics and behavior of the system and can be manipulated to study different aspects of the system.

2. How is parametrization used in scientific research?

Parametrization is used in scientific research to simplify complex systems and make them easier to study and analyze. By breaking down a system into its constituent parameters, scientists can manipulate and control these variables to understand the underlying mechanisms and relationships between them.

3. What is meant by the number of free variables in a system?

The number of free variables in a system refers to the number of independent parameters that can be varied or changed without affecting the overall behavior of the system. These variables are not constrained by other parameters and can be manipulated to observe their effects on the system.

4. How does the number of free variables affect the complexity of a system?

The number of free variables in a system directly affects its complexity. As the number of free variables increases, the system becomes more complex and difficult to analyze. This is because there are more independent factors that can influence the behavior of the system, making it harder to predict and understand.

5. Can parametrization be used in all scientific fields?

Yes, parametrization can be used in all scientific fields. It is a fundamental concept in mathematics and is applied in various fields such as physics, chemistry, biology, and engineering. Parametrization allows scientists to create models and simulations that can be used to study and understand complex systems in these fields.

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