Line Integral vs. Surface Integral: Range of t?

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Discussion Overview

The discussion centers on the differences between line integrals and surface integrals, specifically focusing on the parameterization of a curve and the appropriate range of the parameter \( t \). Participants explore the implications of different parameterizations and how they relate to the endpoints of a curve in three-dimensional space.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants define a line integral as being along a one-dimensional line and a surface integral as over a two-dimensional surface.
  • One participant proposes that the range of \( t \) for the parameterization \( x=t, y=t \) is from 0 to 1, questioning this range.
  • Another participant argues that the range of \( t \) depends on the specific endpoints of the curve, suggesting that if the curve is from \( (x_0, y_0) \) to \( (x_1, y_1) \), the range of \( t \) should correspond to the values that yield these endpoints.
  • A participant presents a specific parameterization \( x=t, y=2t, z=t \) and states that \( t \) should range from 0 to 2, while another reference book states it should be from 0 to 1.
  • One participant clarifies that the parameterization must agree with the endpoints, providing examples of different parameterizations that yield different ranges for \( t \).
  • There is a reiteration of the need to check that the parameterization describes a line, with emphasis on the conditions for \( t \) based on the endpoints in three dimensions.

Areas of Agreement / Disagreement

Participants express differing views on the appropriate range of \( t \) for the given parameterization, with some asserting it should be from 0 to 1 and others suggesting it could be from 0 to 2 depending on the parameterization used. The discussion remains unresolved regarding the correct range of \( t \).

Contextual Notes

Participants reference specific parameterizations and their implications for the range of \( t \), but there is no consensus on the correct interpretation of the parameterization in relation to the endpoints. The discussion highlights the importance of verifying that parameterizations accurately describe the intended curves.

nothGing
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what is the different between line integral and surface integral?
If we parameterize curve by x=t , y=t , what is the range of t ? Is it 0<= t <=1? why?
 
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nothGing said:
what is the different between line integral and surface integral?
A "line integral" is along a one-dimensional line and a "surface integral" is over a two dimensional surface.

If we parameterize curve by x=t , y=t , what is the range of t ? Is it 0<= t <=1? why?
There are no bounds. if you are talking about a portion of a curve, say from (x_0, y_0) to (x_1, y_1), the range of t is from whatever value of t gives x_0 and y_0 to whatever value of t gives x_1 and y_1.
 
Erm.. ok..
Let say, integral ( x^2 + y + z)ds , the line segment of curve is from (0,0,0) to (1,2,1).
We parameterize curve C by x=t , y=2t , z=t.
As you said, t should be between 0 and 2.
but according to my reference book, 0<=t<=1.
Why?
 
HallsofIvy said that the range of t is such that your parametrization will agree with your end points. (Of course, you first need to check that your parametrization does describe a line, in this case it does).

With your paramterization, its 0<=t<=1. But I could choose:

x=t/2, y=t, z=t/2

Which still describe a line with the same orientation, instead now t must range from 0 to 2 to give the desired segment.
 
nothGing said:
Erm.. ok..
Let say, integral ( x^2 + y + z)ds , the line segment of curve is from (0,0,0) to (1,2,1).
We parameterize curve C by x=t , y=2t , z=t.
As you said, t should be between 0 and 2.
but according to my reference book, 0<=t<=1.
Why?
When t=1, x,y,z is the final endpoint. Try plugging t=1 into the forumulas for x,y,z to convince yourself of this.
 
nothGing said:
Erm.. ok..
Let say, integral ( x^2 + y + z)ds , the line segment of curve is from (0,0,0) to (1,2,1).
We parameterize curve C by x=t , y=2t , z=t.
As you said, t should be between 0 and 2.
No, I did NOT say that! I said " if you are talking about a portion of a curve, say from (x0, y0) to (x0, y0) , the range of t is from whatever value of t gives x0 and y0 to whatever value of t gives x1 and y1" . Since this is in three dimensions, we need to include z0 and z1 also.

We must have x(t)= t= 0, y(t)= 2t= 0, and z(t)= t= 0. Obviously t= 0 satisfies all of those.

We must also have x(t)= t= 1, y(t)= 2t= 2, and z(t)= t= 1. Obviously t= 1, NOT t= 2, satisfies all of those.

but according to my reference book, 0<=t<=1.
Why?
 
HallsofIvy, ya.. i missunderstand already.. :p
Well, now i understand it.. thanks to elibj123,Redbelly98, HallsofIvy for helping..
 

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