13.3 – Arc Length and Curvature
•
The curvature of a curve can be defined several ways
•
Be able to reparameterize a curve in terms of the arc length parameter
o
=
+
+
st
0tdxdu2
dydu2
dzdu2du
o
Do the integral, and solve t in terms of s, then substitute into r(t)
•
The curvature can be determined by all of the following ways
o
=
=
( )
( )=
( )
( )
κt
dTds
T' t r' t
r't x
r'' t r' t 3
o
It may be hard to memorize, but try and think of it as steps
o
The individual steps are not hard to calculate, just take it one step at a
time
o
Remember you must calculate the derivatives before plugging in the
point, however after that it may simplify calculation to plug in for t after
that
o
=
(
)
+
(
)
/
κx
f'' x 1
f' x 23 2
o
Don’t worry too much about this formula, but you may want to look at a
couple of examples.
•
Extra Exercises: pg 869 #19, #21, #23,
o
14.1Functions of Several Variables
•
Remember how to solve for the domain of a function.
It’s the same way as in
calc 1 just with more than two variables.
•
Recall the definition of level curves, on pg 892
•
Extra Exercises: pg 898 #7, #13, #19, #23,
#25
o
14.2Limits and Continuity
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•
It is no longer enough to say a limit exists if you approach it from the left and
the right
•
You must prove that it will be the same for any path into some point (x,y)
•
The limit laws still exist as in calculus 1, including the squeeze theorem
•
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 Spring '08
 Cox
 Calculus, Arc Length, Derivative, Limit of a function, Gradient, extra exercises

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