Solving the Curvature Problem for Curve C

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Homework Help Overview

The problem involves determining the curvature of a curve defined by a polar equation r=f(theta). The original poster presents a formula for curvature and seeks to understand how to apply it using derivatives of the curve's representation in Cartesian coordinates.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the need to compute the first and second derivatives of the curve's representation and how to correctly substitute them into the curvature formula. There is confusion regarding the representation of the curve as a vector and the implications of the polar form.

Discussion Status

Participants are actively engaging with the problem, with some providing guidance on the vector representation of the curve. There is recognition of the complexity involved in substituting the derivatives into the curvature formula, and multiple interpretations of the curvature expression are being explored.

Contextual Notes

Some participants express uncertainty about the substitution process and the relationship between different forms of the curvature formula. There is mention of potential confusion arising from the notation used in the problem statement.

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Homework Statement



A Curve C is given by the polar equation r=f(theta). Show that the curvature K at the point (r, theta) is
K=|2(r')^2 - rr'' + r^2|
--------------------
[(r')^2 + r^2]^(3/2)

*Represent the curve by r(theta) = r<cos theta, sin theta>

Homework Equations



I have so far taken the first and second derivatives of x= r cos Theta and y=r sin theta
and I know that the formula below is probably involved but i don't know how
K(t) = ||r'(t) X r''(t)||
---------------
||r'(t)||^3
 
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Write your second formula K(theta)=|f'(theta) X f''(theta)|/|f'(theta)|^3 where f(theta)=r(theta)*[cos(theta),sin(theta)]. Now put your derivatives of f(theta) in. Your presentation above has too many r's in it. Is that what's confusing?
 
yea, i dnt know exactly what I'm doing when I am plugging things in.. like f(theta) is a vector?.. f(theta) = r cos theta + r sin theta... or... I just don't understand :( but i get the math, just not what I am subsituting..
 
f(theta) IS a vector [r(theta)*cos(theta),r(theta)*sin(theta)]. Those are the x and y components. Consider theta a parameter of the curve, like t.
 
i got that the actual K should be 1/ (r^2 + 1)^(1/2) which i can't seem to relate to the K given.. when i subsitute it, it's too messy
 
multivariable said:
i got that the actual K should be 1/ (r^2 + 1)^(1/2) which i can't seem to relate to the K given.. when i subsitute it, it's too messy

How did you get that? I thought you were going to (or had) computed f'(theta) and f''(theta). That's the right way to go. Start with that.
 

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