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Math Help - Curvature problem

  1. #1
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    Curvature problem

    Let k(t),  t \geq 0 construct the curve a(t), for which the curvature at t \geq 0 is equal to k(t) and which satisfies the initial conditions a(0) = (0,0) and a'(0) = (1,0).

    Prove that  a(t) = ( \int ^{t}_{0} cos ( \int ^{s}_{0}k(u)du)ds , \int ^{t}_{0} sin ( \int ^{s} _{0} k(u)du) ds )

    Questions:

    Now, if I substitute 0 in for t, I get (0,0). But taking the prime, should I take it relative to t or s?
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  2. #2
    Super Member Rebesques's Avatar
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    Just to get you going.

    Let the curve be c(t)=(x(t),y(t)), t\geq 0. Suppose (without loss of generality) c is already parametrized by arc length. Then the formula |c'(t)|=1 gives x'(t)^2+y'(t)^2=1, so there exists a function \phi(t) such that x'(t)=\cos(\phi(t)), \ y'(t)=\sin(\phi(t)). To explicitly calculate this function, recall that k(t)=x'(t)y''(t)-x''(t)y'(t).
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