Stewart Calculus 13.3

1. 13.3 Arc Length and
Curvature
2D Arc Length Formula:
= [ ( )] + [ ( )]

2. 13.3 Arc Length and
Curvature
2D Arc Length Formula:
= [ ( )] + [ ( )]
3D Arc Length Formula:
= [ ( )] + [ ( )] + [ ( )]

3. 13.3 Arc Length and
Curvature
2D Arc Length Formula:
= [ ( )] + [ ( )]
3D Arc Length Formula:
= [ ( )] + [ ( )] + [ ( )]
Vector Form
= | ( )|

4. There are in general several parametrizations
of a single curve.
Ex: ()= , , ,
( )= ,( ) ,( ) ,
( )= , , ,
they all represent the same curve.
There is one parametrization that is special.

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 can
parametrize the curve with respect to :
= ( ( ))

6. Ex: Reparametrize the helix curve ( ) = , ,
with respect to arc length measured from ( , , ).
When = , ( ) = ( , , ).
= | ( )| =
()= | ( )| = =
= /
( ( )) = , ,

7. The unit tangent vector of a curve () is
defined as:
()
()=
| ( )|
The curvature of a curve is
=
curvature is small
curvature is large

8. Alternative formulae:
| ( )| | () ( )|
= = =
| ( )| | ( )|
Ex: Find the curvature of ()= , , at ( , , ).

9. Alternative formulae:
| ( )| | () ( )|
= = =
| ( )| | ( )|
Ex: Find the curvature of ()= , , at ( , , ).
()= , , , ()= , ,

10. Alternative formulae:
| ( )| | () ( )|
= = =
| ( )| | ( )|
Ex: Find the curvature of ()= , , at ( , , ).
()= , , , ()= , ,
() ()= = , ,

11. Alternative formulae:
| ( )| | () ( )|
= = =
| ( )| | ( )|
Ex: Find the curvature of ()= , , at ( , , ).
()= , , , ()= , ,
() ()= = , ,
| , , | + +
()= = /
| , , | ( + + )

12. Alternative formulae:
| ( )| | () ( )|
= = =
| ( )| | ( )|
Ex: Find the curvature of ()= , , at ( , , ).
()= , , , ()= , ,
() ()= = , ,
| , , | + +
()= = /
| , , | ( + + )
( )=

13. Ex: Show that the curvature of a circle of
radius is / everywhere.

14. Ex: Show that the curvature of a circle of
radius is / everywhere.
A circle can be represented as
()= , ,

15. Ex: Show that the curvature of a circle of
radius is / everywhere.
A circle can be represented as
()= , ,
()= , ,

16. Ex: Show that the curvature of a circle of
radius is / everywhere.
A circle can be represented as
()= , ,
()= , ,
()
()= = , ,
| ( )|

17. Ex: Show that the curvature of a circle of
radius is / everywhere.
A circle can be represented as
()= , ,
()= , ,
()
()= = , ,
| ( )|
()= , ,

18. Ex: Show that the curvature of a circle of
radius is / everywhere.
A circle can be represented as
()= , ,
()= , ,
()
()= = , ,
| ( )|
()= , ,
| ( )|
()= =
| ( )|

19. Ex: Find the curvature of the parabola =
at the point ( , ).

20. Ex: Find the curvature of the parabola =
at the point ( , ).
The curve can be parametrized as ( )= , , .

21. Ex: Find the curvature of the parabola =
at the point ( , ).
The curve can be parametrized as ( )= , , .
( )= , , . ( )= , , .

22. Ex: Find the curvature of the parabola =
at the point ( , ).
The curve can be parametrized as ( )= , , .
( )= , , . ( )= , , .
( ) ( )= , ,

23. Ex: Find the curvature of the parabola =
at the point ( , ).
The curve can be parametrized as ( )= , , .
( )= , , . ( )= , , .
( ) ( )= , ,
( )= /
( + )

24. Ex: Find the curvature of the parabola =
at the point ( , ).
The curve can be parametrized as ( )= , , .
( )= , , . ( )= , , .
( ) ( )= , ,
( )= /
( + )
( )= .

25. Ex: Find the curvature of the parabola =
at the point ( , ).
The curve can be parametrized as ( )= , , .
( )= , , . ( )= , , .
( ) ( )= , ,
( )= /
( + )
( )= .
In general a plane curve = ( ) has curvature
( )
( )= /
[ + ( ( )) ]