How Should I Calculate Curvature for Standard and Vector-Valued Functions?

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SUMMARY

The discussion centers on calculating curvature for the functions y=cos(x) and y=e^x using vector-valued functions. The user initially attempted to express y=cos(x) as r(t) = and applied the curvature formula |v(t) x a(t)| / |v(t)|^3. However, they expressed uncertainty about the validity of their approach, particularly when considering alternative vector functions like . Additionally, they encountered imaginary roots while trying to determine the maximum curvature of y=e^x, leading to confusion about the concept of curvature itself. A suggestion was made to refer to a specific curvature formula from MathWorld, indicating that a three-dimensional approach may not be necessary.

PREREQUISITES
  • Understanding of curvature in calculus
  • Familiarity with vector-valued functions
  • Knowledge of cross products in vector calculus
  • Basic concepts of limits and continuity in functions
NEXT STEPS
  • Study the curvature formula for planar curves as outlined in MathWorld, specifically equation 14.
  • Learn how to derive curvature for functions without converting them to vector-valued forms.
  • Explore the implications of using different parameterizations for vector-valued functions.
  • Investigate the behavior of exponential functions and their curvature properties as t approaches infinity.
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Students studying calculus, particularly those focusing on curvature, vector-valued functions, and their applications in mathematical analysis.

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Homework Statement


For the first problem I am asked to find the curvature for y=cosx

We are studying vector value functions so I tried to rewrite this as a vector valued function so I can find the curvature. I just chose r(t)= <t,cost,0>. I found rI(t)=<1,-sint,0> and rII(t)=<0,-cost,0> and used the curvature formula of lv(t)xa(t)l/(lv(t)l^3 to find the curvature...but then I thought that if I had used a different vector valued function such as <t^3,cost^3,0> it would not work out the same so I am not sure how I should tackle this problem now.Second Problem is similar...It is asking "at what point does y= e^x have the maximum curvature" so I tried doing a similar approach as I did in the previous problem (even though I suspect my method is faulty) and got imaginary roots when solving for a point which makes me think the maximum curvature might be as t->infinity but I'm not sure.

I'm having difficulty with the concept of curvature here I think, especially because the function is not already written as a vector valued function (maybe their is a way to deal with this without vector valued functions, I'm not sure.)

Homework Equations

The Attempt at a Solution


Sorry kind of attempted solution in the 1st part...

Thank you[/B]
 
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Austin said:

Homework Statement


For the first problem I am asked to find the curvature for y=cosx

We are studying vector value functions so I tried to rewrite this as a vector valued function so I can find the curvature. I just chose r(t)= <t,cost,0>. I found rI(t)=<1,-sint,0> and rII(t)=<0,-cost,0> and used the curvature formula of lv(t)xa(t)l/(lv(t)l^3 to find the curvature...but then I thought that if I had used a different vector valued function such as <t^3,cost^3,0> it would not work out the same so I am not sure how I should tackle this problem now.Second Problem is similar...It is asking "at what point does y= e^x have the maximum curvature" so I tried doing a similar approach as I did in the previous problem (even though I suspect my method is faulty) and got imaginary roots when solving for a point which makes me think the maximum curvature might be as t->infinity but I'm not sure.

I'm having difficulty with the concept of curvature here I think, especially because the function is not already written as a vector valued function (maybe their is a way to deal with this without vector valued functions, I'm not sure.)

Homework Equations

The Attempt at a Solution


Sorry kind of attempted solution in the 1st part...

Thank you[/B]

You don't have to promote the problem to a three dimensional vector problem. Look at equation 14) here. http://mathworld.wolfram.com/Curvature.html
 
Dick said:
You don't have to promote the problem to a three dimensional vector problem. Look at equation 14) here. http://mathworld.wolfram.com/Curvature.html
Oh, thanks! Didn't realize there was such a formula. We've just been studying vector valued functions so putting it into 3d and using that formula was all i knew
 

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