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Curvature problem

  • #1

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

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
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618
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?
 
  • #3
yea, i dnt know exactly what i'm doing when im 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 im subsituting..
 
  • #4
Dick
Science Advisor
Homework Helper
26,258
618
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.
 
  • #5
i got that the actual K should be 1/ (r^2 + 1)^(1/2) which i cant seem to relate to the K given.. when i subsitute it, it's too messy
 
  • #6
Dick
Science Advisor
Homework Helper
26,258
618
i got that the actual K should be 1/ (r^2 + 1)^(1/2) which i cant 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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