Defining implicit function given a parametric function

In summary, the problem is to define a continuous implicit function g : R3 -> R corresponding to a 2D surface in 3D space described by a continuous parametric function f : R2 -> R3. The solution involves using the infimum function and considering g as the distance from a point to the surface, which can be defined in terms of an infimum.
  • #1
mathnewbie123
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Homework Statement




Given a continuous parametric function f : R2 -> R3 specifying a 2D
surface in 3D space, define a continuous implicit function g : R3 -> R
corresponding to the same surface.

Homework Equations



You’ll likely want to use the infimum function.
You can ignore the inside/outside convention: g can be everywhere
nonnegative.

The Attempt at a Solution



My thought on this is that... Since g is an implicit function and is non negative everywhere else.. Then if a point lies on the surface described by g, g(x,y,z) = 0 at that point. Since g is zero on the surface and continuous and increasing everywhere else, it must include an infimum of some sort and a modified function of f, where f(s,t) = (x,y,z).. And I'm stuck from then on..

Someone help please? Any help is appreciated..
 
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  • #2
I think they are trying to get you to think of defining g as the distance from a point to the surface. Can you define this distance in terms of an infimum?
 

FAQ: Defining implicit function given a parametric function

1. What is an implicit function?

An implicit function is a mathematical relationship between variables that is not explicitly stated. This means that one variable is defined in terms of the other, rather than both variables being explicitly defined.

2. What is a parametric function?

A parametric function is a mathematical expression that defines a relationship between two or more variables, where each variable is represented by a parameter. The parameters can take on different values, resulting in different values for the function.

3. How do you define an implicit function given a parametric function?

To define an implicit function given a parametric function, you need to eliminate the parameter(s) by solving for one variable in terms of the other. This will result in an equation that relates the two variables without explicitly stating one in terms of the other.

4. What is the benefit of defining an implicit function?

Defining an implicit function allows for a more general representation of a relationship between variables. It can also make it easier to solve for one variable in terms of the other, as the equation is typically simpler than the parametric function.

5. How is an implicit function different from an explicit function?

An explicit function explicitly states one variable in terms of the other, while an implicit function does not. This means that an explicit function can be easily solved for one variable, while an implicit function requires more steps to solve for one variable in terms of the other.

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