13.3 Arc Length and     Curvature2D Arc Length Formula:        =      [ ( )] + [ ( )]
13.3 Arc Length and     Curvature2D Arc Length Formula:        =       [ ( )] + [ ( )]3D Arc Length Formula:   =        [ ...
13.3 Arc Length and     Curvature2D Arc Length Formula:        =       [ ( )] + [ ( )]3D Arc Length Formula:   =        [ ...
There are in general several parametrizationsof a single curve.Ex:          ()=   ,        ,   ,      ( )=             ,( ...
We define the arc length function   ( ) by            ()=     | ( )|Using the Fundamental Theorem of Calculus,we have     ...
Ex: Reparametrize the helix curve ( ) =       ,       ,with respect to arc length measured from   ( , , ).When    =   ,   ...
The unit tangent vector of a curve    ()   isdefined as:                     ()               ()=                   | ( )|...
Alternative formulae:                    | ( )|   | ()      ( )|          =       =        =                    | ( )|    ...
Alternative formulae:                      | ( )|   | ()      ( )|          =         =        =                      | ( ...
Alternative formulae:                        | ( )|   | ()      ( )|            =         =        =                      ...
Alternative formulae:                        | ( )|   | ()      ( )|            =         =        =                      ...
Alternative formulae:                        | ( )|   | ()      ( )|            =         =        =                      ...
Ex: Show that the curvature of a circle ofradius is / everywhere.
Ex: Show that the curvature of a circle ofradius is / everywhere.A circle can be represented as          ()=         ,    ...
Ex: Show that the curvature of a circle ofradius is / everywhere.A circle can be represented as          ()=         ,    ...
Ex: Show that the curvature of a circle ofradius is / everywhere.A circle can be represented as          ()=         ,    ...
Ex: Show that the curvature of a circle ofradius is / everywhere.A circle can be represented as          ()=         ,    ...
Ex: Show that the curvature of a circle ofradius is / everywhere.A circle can be represented as          ()=         ,    ...
Ex: Find the curvature of the parabola   =at the point ( , ).
Ex: Find the curvature of the parabola    =at the point ( , ).The curve can be parametrized as   ( )=   ,   ,   .
Ex: Find the curvature of the parabola    =at the point ( , ).The curve can be parametrized as   ( )=   ,   ,   .    ( )= ...
Ex: Find the curvature of the parabola       =at the point ( , ).The curve can be parametrized as     ( )=    ,   ,   .   ...
Ex: Find the curvature of the parabola         =at the point ( , ).The curve can be parametrized as       ( )=    ,   ,   ...
Ex: Find the curvature of the parabola          =at the point ( , ).The curve can be parametrized as        ( )=    ,   , ...
Ex: Find the curvature of the parabola                =at the point ( , ).The curve can be parametrized as            ( )=...
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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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