Parametrizing Paths: Learn How to Do It for Functions of Multiple Variables

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In summary, "parametrization" refers to a function that maps a curve or path in n-dimensional space to a one-dimensional object. There are various methods for parameterizing a path, including using the function itself, arclength, physical motion, and geometric properties. It is important to note that there is no single "correct" parameterization for any path, as there are infinite possibilities.
  • #1
nivekious
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I'm trying to figure out what is meant by parametrizing a path, and how it would be done for a function of multiple variables. Can someone help me?
 
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  • #2
If you mean by path a curve C in, say R^n, then "parametrization" usually refers to a function

[tex]
\gamma:[a,b]\to\mathbb{R}^n
[/tex]

such that [itex]\gamma([a,b])=C[/itex].
This is just the rough idea, does it help?
 
  • #3
not meant to derail but what are some techniques for parameterizing arbitrary paths? probably interpolation?
 
  • #4
Since "interpolation" is an approximating method, no it is not at all appropriate for a problem like this.

The whole point of parameterization is that a path, whether in 2 dimensional space or 3 dimensional space, is a one dimensional object. We should be able to specify any point on it by one number.

Another important thing to remember is that there is no single "correct" parameterization for any path- there always exist an infinite number of such parameters.

One fairly useful "method" is this: if y= f(x) is a function of x then we can use x itself as parameter: x= x and y= f(x) or, if you prefer to call your parameter "t", x= t, y= f(t).

It is always possible to use "arclength" as parameter: select some point on the path as t= 0 and one direction as t> 0 (both of those can be done arbitrarily). The (x,y) corresponding to t> 0 is the point at distance t from the point at t=0 in that direction, and the (x,y) corresponding to t< 0 is the point at distance t from the point at t=0 in the opposite direction. Of course, determining formulas for x and y, as functions of t involves doing a (typically) difficult integral to find the arclength itself- I suspect you haven't done that yet.

You can think "physically": imagine an object moving along the path with a given speed. Then (x(t), y(t)) is the point your object is at at time t. That's often done, of course, in Physics.

Finally, you can use some kind of geometric property. I know that, if I measure angle [itex]\theta[/itex] from the positive x axis, the point on a circle of radius R, center at the origin, at angle [itex]\theta[/itex] is given by (Rcos([itex]\theta[/itex]), Rsin([itex]\theta[/itex]) so I can use x= Rcos([itex]\theta[/itex]), y= Rsin([itex]\theta[/itex]) as parametric equations for that circle.

The important thing to remember is that there is no single "correct" parameterization: every path has an infinite number of possible parameterizations.
 

1. What is parametrization?

Parametrization is the process of representing a curve or surface in a higher dimensional space by using a set of parameters. It allows us to describe the path of a function of multiple variables in terms of a single variable.

2. Why is parametrization useful?

Parametrization is useful because it simplifies the representation of complex functions of multiple variables. By using a set of parameters, we can break down the path into smaller, more manageable segments, making it easier to analyze and manipulate the function.

3. How do you parametrize a path for a function of multiple variables?

To parametrize a path for a function of multiple variables, we need to find a set of parameters that can describe the path. This can be achieved by setting up a system of equations that relate the parameters to the variables of the function. Once we have the parameters, we can use them to represent the path and manipulate the function as needed.

4. What are the benefits of parametrizing paths?

Parametrizing paths allows us to easily manipulate and analyze complex functions of multiple variables. It also helps us to visualize the path in a higher dimensional space, making it easier to understand the behavior of the function. Additionally, parametrization can be useful in optimization problems, as it allows us to find the minimum or maximum values along the path.

5. What are some common techniques for parametrizing paths?

Some common techniques for parametrizing paths include using trigonometric functions, polynomial functions, or rational functions as parameters. These functions can be manipulated to fit the specific path and function being studied. Another technique is to use a change of variables, where we replace the original variables with new ones that are easier to work with in terms of the parameters.

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