Parametrization of a curve

In summary, the problem involves finding a parametrization of a curve that follows a corkscrew pattern, starting at (\sqrt{2\pi},0,0) and ending at (0,0,2\pi), along the surface defined by z= 2\pi- x^2- y^2. The attempt at a solution involved using x=r \cos t and y=r \sin t, but the problem lies in finding the correct parametrization for the curve.
  • #1
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Homework Statement



I'm doing a line integral and can't seem to figure out the parametrization of this curve:
[itex]x^2+y^2+z=2\pi[/itex]

Homework Equations


Looking to get it to the form:
[itex]\textbf{c}(r,t)=(x(r,t),y(r,t),z(r,t))[/itex] (I don't even know if this is right though).

The Attempt at a Solution


Trying to use [itex]x=r \cos t[/itex] and [itex]y=r \sin t[/itex] but I still can't get anywhere.

I have a feeling I'm totally in the wrong direction.
The [itex]2\pi[/itex] is killing me too!
 
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  • #2
Your basic problem is NOT the "[itex]2\pi[/itex]". It is that [itex]x^2+ y^2+ z= 2\pi[/itex] does NOT define a line (or curve or path) in three dimensions. It can be written as [itex]z= 2\pi- x^2- y^2[/itex] which is a surface (specifically, a paraboloid). Essentially, given any x and y you can solve for z so this is a two dimensional figure, not one dimension.

Please tell us what the entire problem really is.
 
  • #3
I realized this after thinking about for a while, the real parametrizaion I can't figure out is a curve that is a corkscrew getting narrower as it goes up around the parabolioid, starting at [itex] (\sqrt{2\pi},0,0) [/itex] and ending at the top of the paraboloid, [itex] (0,0,2\pi) [/itex]
 

1. What is the purpose of parametrizing a curve?

The purpose of parametrizing a curve is to represent the curve as a set of equations in terms of a parameter. This allows for easier calculation and analysis of the curve's properties, such as its length, slope, and curvature.

2. How is a curve parametrized?

A curve can be parametrized by expressing its coordinates in terms of a parameter, typically denoted as t. This parameter can represent time, distance, or any other variable that helps define the curve.

3. What are the advantages of using a parametrized curve?

Using a parametrized curve allows for more flexibility in representing and analyzing the curve. It also simplifies the process of finding derivatives and integrals of the curve, as well as calculating arc length and surface area.

4. Can any curve be parametrized?

Yes, any curve can be parametrized as long as it is continuous and has a one-to-one correspondence between the parameter and the points on the curve. However, some curves may have more complex parametrizations than others.

5. How does parametrization affect the orientation of a curve?

The orientation of a curve can be affected by the direction of the parameter. If the parameter increases in the same direction as the curve, the orientation remains the same. However, if the parameter increases in the opposite direction, the orientation of the curve will be reversed.

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