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Math Help - circumference of a cirlce

  1. #1
    Ant
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    circumference of a cirlce

    I'm definitely doing something wrong here, just not sure what!

    Let \gamma be the anticlockwise oriented circle centered at z_{o} \in \mathbb{C} of radius r.

    Clearly (from the formula for the circumference of a circle) | \gamma | = 2\pi r .

    Using formula though:

    | \gamma | = \int_a^b | \gamma'(t) | dt .

    Suppose we parameterize \gamma as follows:

    \gamma(t)=z_{0} + re^{it} \ t \in [0,2\pi]

    Hence  \gamma'(t) = tre^{it}

    And so,

    | \gamma | = \int_a^b | \gamma'(t) | dt =\int_0^{2\pi} |tre^{it}| \ dt = r \int_0^{2\pi} t \ dt = r(\frac{4\pi^2}{2} - 0) = 2\pi^2 r \ne 2\pi r

    Can anyone tell me what I'm doing wrong? Thanks!
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  2. #2
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    Re: circumference of a cirlce

    the derivative is re^it*i
    Thanks from Ant and Ruun
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  3. #3
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    Re: circumference of a cirlce

    Indeed, the parameter is t so the derivative shall be with respect to t
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