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Calculus II - 36

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Stewart Calculus 13.3

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Calculus II - 36

  1. 1. 13.3 Arc Length and Curvature2D Arc Length Formula: = [ ( )] + [ ( )]
  2. 2. 13.3 Arc Length and Curvature2D Arc Length Formula: = [ ( )] + [ ( )]3D Arc Length Formula: = [ ( )] + [ ( )] + [ ( )]
  3. 3. 13.3 Arc Length and Curvature2D Arc Length Formula: = [ ( )] + [ ( )]3D Arc Length Formula: = [ ( )] + [ ( )] + [ ( )]Vector Form = | ( )|
  4. 4. There are in general several parametrizationsof a single curve.Ex: ()= , , , ( )= ,( ) ,( ) , ( )= , , ,they all represent the same curve.There is one parametrization that is special.
  5. 5. We define the arc length function ( ) by ()= | ( )|Using the Fundamental Theorem of Calculus,we have = | ( )|If we can solve as a function of , we canparametrize the curve with respect to : = ( ( ))
  6. 6. Ex: Reparametrize the helix curve ( ) = , ,with respect to arc length measured from ( , , ).When = , ( ) = ( , , ). = | ( )| = ()= | ( )| = = = / ( ( )) = , ,
  7. 7. The unit tangent vector of a curve () isdefined as: () ()= | ( )|The curvature of a curve is = curvature is small curvature is large
  8. 8. Alternative formulae: | ( )| | () ( )| = = = | ( )| | ( )|Ex: Find the curvature of ()= , , at ( , , ).
  9. 9. Alternative formulae: | ( )| | () ( )| = = = | ( )| | ( )|Ex: Find the curvature of ()= , , at ( , , ). ()= , , , ()= , ,
  10. 10. Alternative formulae: | ( )| | () ( )| = = = | ( )| | ( )|Ex: Find the curvature of ()= , , at ( , , ). ()= , , , ()= , , () ()= = , ,
  11. 11. Alternative formulae: | ( )| | () ( )| = = = | ( )| | ( )|Ex: Find the curvature of ()= , , at ( , , ). ()= , , , ()= , , () ()= = , , | , , | + + ()= = / | , , | ( + + )
  12. 12. Alternative formulae: | ( )| | () ( )| = = = | ( )| | ( )|Ex: Find the curvature of ()= , , at ( , , ). ()= , , , ()= , , () ()= = , , | , , | + + ()= = / | , , | ( + + ) ( )=
  13. 13. Ex: Show that the curvature of a circle ofradius is / everywhere.
  14. 14. Ex: Show that the curvature of a circle ofradius is / everywhere.A circle can be represented as ()= , ,
  15. 15. Ex: Show that the curvature of a circle ofradius is / everywhere.A circle can be represented as ()= , , ()= , ,
  16. 16. Ex: Show that the curvature of a circle ofradius is / everywhere.A circle can be represented as ()= , , ()= , , () ()= = , , | ( )|
  17. 17. Ex: Show that the curvature of a circle ofradius is / everywhere.A circle can be represented as ()= , , ()= , , () ()= = , , | ( )| ()= , ,
  18. 18. Ex: Show that the curvature of a circle ofradius is / everywhere.A circle can be represented as ()= , , ()= , , () ()= = , , | ( )| ()= , , | ( )| ()= = | ( )|
  19. 19. Ex: Find the curvature of the parabola =at the point ( , ).
  20. 20. Ex: Find the curvature of the parabola =at the point ( , ).The curve can be parametrized as ( )= , , .
  21. 21. Ex: Find the curvature of the parabola =at the point ( , ).The curve can be parametrized as ( )= , , . ( )= , , . ( )= , , .
  22. 22. Ex: Find the curvature of the parabola =at the point ( , ).The curve can be parametrized as ( )= , , . ( )= , , . ( )= , , . ( ) ( )= , ,
  23. 23. Ex: Find the curvature of the parabola =at the point ( , ).The curve can be parametrized as ( )= , , . ( )= , , . ( )= , , . ( ) ( )= , , ( )= / ( + )
  24. 24. Ex: Find the curvature of the parabola =at the point ( , ).The curve can be parametrized as ( )= , , . ( )= , , . ( )= , , . ( ) ( )= , , ( )= / ( + ) ( )= .
  25. 25. Ex: Find the curvature of the parabola =at the point ( , ).The curve can be parametrized as ( )= , , . ( )= , , . ( )= , , . ( ) ( )= , , ( )= / ( + ) ( )= .In general a plane curve = ( ) has curvature ( ) ( )= / [ + ( ( )) ]

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