[Return HW; hand out time sheets]
Prof. Tibor Beke will substitute for me on Thursday and
Friday of this week.
Another perspective on the “
v
′
(
t
) = 
v
(
t
)
′
for all
t
”
problem we discussed last time: What values can the left
hand side take?
What about the right hand side?
Section 10.8: Arc length and curvature
Three formulas for the curvature
κ
of a space curve:
κ
= 
d
T
/
ds
 (the definition; requires the arclength
parametrization)
κ
(
t
) = 
T
′
(
t
)/
r
′
(
t
) (any parametrization will do)
κ
(
t
) = 
r
′
(
t
)
×
r
′′
(
t
) / 
r
′
(
t
)
3
An intuitive description of
κ
= 
T
′
(
t
)/
r
′
(
t
):
Since the length of
T
(
t
) is constant, 
T
′
(
t
) is the rate of
change of direction, or turning, of the unit tangent, and 
r
′
(
t
) measures the speed along the curve.
So
κ
is the rate of turning
of the unit tangent divided by the
speed
along the curve.
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For a circle in the plane parametrized in the standard way,
r
(
t
) =
⟨
a
cos
t
,
a
sin
t
⟩
(with
a
> 0), the angle of the unit
tangent will change by 2
π
as we go around the circle (a
total of 2
π
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 Fall '11
 Staff
 Velocity, Osculating circle, Prof. Tibor Beke

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