Multivariable Calculus + Calc III Curvature Notes

10 26.22
LECTURE 27
curving of a curve is reflected by changing direction
of a tangent vector rate of changeof tan vector
Definition if C is a curve and rls is an arc length
parameterization the curvature K s
K S 11 T s II
ble vis is an arc length param 11 T s 11 11 r s 11
K S does not depend on the arc length param
measures rate of changeof T s wrt are length
how does Tls change when arc length
is constant is constant
but curvature is
much different
theorem ri t is a smoothparam of C then suppose
P to and P lies on C then
kit to ftp t both
to calculate
theorem kit r t x r t Il WII11 r t 113

Example compute Ket for ret Acoset Asin t 5
for OETE 2T A O
KA HT t 1
r t I T t R'Ct
Ur t I
r t f Asint Acost O
118 t IF Asin't A'cost 02 A
T t Sint cost O
IT t I Finttost
Ket At big radius circleshave small curvature
LECTURE 27 PROBLEMS
1 B
2 Kit of ret 3 Sint Boost 4t
kit TI T t 3cost Sint 4Ur till
r t 3cost 3sint 4 A
Hr f 11
att3Étt42 5

3 r t Egcoslett sin et gas et
Mr A IF
gasket sin'let 965kt
Ur t 1 1
T t r t and IIT t 1 1
Ket I
4 A