Covering space of implicit vs parametric functions

In summary, the conversation discusses the relationship between implicit functions and parametric functions. The set of implicit functions covers a certain function-space, but it is unclear if every implicit function can be converted to a parametric form or vice versa. The set of implicit functions may not necessarily be a subset of parametric functions. The conversation also mentions the concept of "genus" in relation to the existence of a parameterization for an implicit function. Additionally, the conversation addresses the possibility of converting parametric functions to implicit form and the applicability of these concepts to all sets, not just R^2.
  • #1
ellipsis
158
24
Hello PF, I've got a curiosity question someone may be able to indulge me on:

The set of implicit functions covers a certain function-space - the set of all functions that can be represented by an implicit relation. Parametric functions also covers a function-space, that at least overlaps implicit functions in some points.

Is it the case that every implicit function can be converted to a parametric form? Or are there implicit functions that cannot be represented as a parametric function, and vice versa? Is the set of implicit functions a subset of parametric functions?
 
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  • #2
How exactly do you define "set of implicit functions" and "set of parametric functions"?

As an example, you can always define U(x,y) to be zero for an arbitrary subset of R^2, then the implicit function U(x,y)=0 can be everything.
 
  • #3
I don't have the vocabulary to define these things the proper way. There are functions which just can't be written explicitly in an analytic, closed form. For a quick example, the solution of y in x=y^(y+1). Likewise, there are parametric functions that cannot be written in closed form, such as y=sin(t), x = cos(t). But that parametric function can be converted to implicit form - x^2+y^2 = 1.

Are you saying that the implicit function U(x,y) = 0 is a way of describing any function? I think there might be a function you can write parametrically but not implicitly e.g. x(t) = sin(t)+t, y(t) = cos(t)+t^2

Usually to go from parametric -> implicit, I would solve for T and substitute, but T can be unsolvable for both.

EDIT: I have discovered some information on the topic here. Whether or not an implicit function has a parameterization seems to be related to its 'genus'. Whatever that is.
 
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  • #4
ellipsis said:
Are you saying that the implicit function U(x,y) = 0 is a way of describing any function?
If you do not put additional constraints on U (continuous, or even differentiable, ...), yes.
 
  • #5
So the answer to my question: Any parametric function can be described as an implicit function U(x,y), but not every implicit function can be described as a parametric one.
 
  • #6
You can do the same with parametric functions. x=U(t), y=V(t). As long as you do not add more requirements for U and V, this can give every possible function.
This is not limited to R^2, of course, it works for all sets.
 
  • #7
Try looking at the implicit function theorem, which gives conditions for the existence of a(n) (at least) local representation f(x,y)=0 for a given function.
 

What is the difference between implicit and parametric functions?

Implicit functions are equations that define a relationship between two or more variables, without explicitly giving one variable in terms of the others. Parametric functions, on the other hand, are equations that define each variable in terms of an independent parameter. In other words, implicit functions describe a relationship between variables, while parametric functions describe how each variable changes with respect to a parameter.

How do you find the covering space of an implicit function?

To find the covering space of an implicit function, you can use the Implicit Function Theorem. This theorem states that if an implicit function can be written in the form F(x,y) = 0, where F is a differentiable function, then there exists a unique solution for y in terms of x in a small neighborhood around a given point. This allows us to determine the covering space by solving for y in terms of x.

How is the covering space of a parametric function different from an implicit function?

The covering space of a parametric function is typically easier to find compared to an implicit function. This is because parametric functions explicitly define each variable in terms of an independent parameter, making it easier to manipulate and solve for the covering space. Implicit functions, on the other hand, require the use of the Implicit Function Theorem to determine the covering space.

What are some applications of covering space of implicit and parametric functions?

Covering space is a fundamental concept in calculus and is used in various applications such as optimization problems, finding critical points, and curve sketching. It is also used in physics and engineering to model complex systems and equations that cannot be solved explicitly.

How can understanding covering space of implicit and parametric functions be useful in real life?

Understanding covering space allows us to analyze and solve complex problems in various fields, such as economics, biology, and engineering. It also helps in visualizing and understanding the behavior of functions and their relationships, which can be applied in problem-solving and decision-making processes.

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