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by: Reyes Glover


Reyes Glover
GPA 3.67

Aram Karakhanyan

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About this Document

Aram Karakhanyan
Class Notes
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This 2 page Class Notes was uploaded by Reyes Glover on Sunday September 6, 2015. The Class Notes belongs to M 408M at University of Texas at Austin taught by Aram Karakhanyan in Fall. Since its upload, it has received 54 views. For similar materials see /class/181425/m-408m-university-of-texas-at-austin in Mathematics (M) at University of Texas at Austin.

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Date Created: 09/06/15
Explanation of problem 1 b Here we discuss the problem 1 b In general terms we consider a parametric curve given by z acost and y bsin t Let us rst identify this curve we have that za cost and yb sint therefore 2 2 i coszt sinzt 1 a b hence we have an ellipse Now to compute the arc length we use our formula we assume that a lt b d 2 quot dt a2s1n2tb200s2tdt 01 0 0 7r 7r b2 7 a2 H12 b2 7 12 cos2 tdt a 1 2 cos2 tdt 02 0 0 a a 1 82 cos tdt 03 0 where 8 xb2 7 aZa is the eccentricity of the ellipse Now let us set 2 cost then It arccosz and dt darccost 71x1 7 22 so using change of variable formula we get 1722 7r 1 1 2 2 L a 182cos2tdta idz 04 0 0 Consider the inde nite integral f lff 2dz for 8 gt 0 Rationalizing the numerator one has 1 2 2 1 2 2 8 z La 05 1 2 172216222 1 2 r 1 82 Z r 06 172218222 172218222 Finally let us take 2 V1 7 u7 then dz 7 du This transforms for instance V17722 the rst integral into 1 1 u 172216222 u18217u217u2 1 1 7 7 08 x18217u217k222 where we set k 81 82 Following to Legendrel we call integrals 1 2 rim u rim 0 lt k lt1 09 lt17 u2gtlt17 W lt17 u2gtlt17 W respectively elliptic integrals of lst and 2nd kind This integrals can not be explicitly computed via elementary functions We observe here that for 1st integral in 09 we have a different representation after substitution 2 sin p then it reads dltp W39 This integrals are important in mathematical physics Finally we give two other examples of computation of arc lengths which lead to elliptic integrals Example 1 Compute the arc length of y CS1H 70 gt 0a gt 0 for z E OJa We have that 7ra 2 7r L 111L672cos2 da182cos2zdx7 83 0 a a 0 a which as we have seen reduces to the sum of elliptic integrals of rst and second kind Example 2 Compute the arc length of 77snail given by r acost b In this case it is easy to see that we again get a combination of elliptic integrals of 1st and 2nd kind Aram L Karakhanyan 1AdrienMarie Legendre 1752 18337 French Mathematician


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