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Math 241 Chapter 12

Dr
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Wyss-Gallifent

§12
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Definition: For each t, F (t) (or usually, and later, r(t)) points from the origin to a point on the
¯
curve
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Classic examples: Circles, helices, lines and line segments, functions
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Properties: Without going too far into detail note that VVFs are vectors and so we can do
¯
¯
¯
vectorish things with them like F × G and f F where f is a regular function
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2 Limits and Continuity of VVFs
1
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We can then
define a VVF to be continuous iff the components are continuous
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§12
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We can do this formally with limits but we won’t
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2
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The function s(t) = ||¯(t)|| is speed
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¯
¯
¯
3
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The integral of the VVF is just the integral of the components
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5
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§12
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Definition: A (space) curve is just the curve itself, without worrying initially about the parametrization
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¯
2
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In other words it forms a loop
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¯
¯
0
(c) A curve is piecewise smooth if it can be divided up into a number of smooth pieces
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r

§12
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Defn: The tangent vector is T (t) = ||¯′ (t)||
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Also note that T (t) is a unit vector
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Defn: The normal vector is N (t) = ||T ′ (t)||
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If the curve does
¯
not bend then the normal vector does not exist
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6 Curvature
¯′

(t)||
1
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Another way to get it is κ(t) =
||¯
r
formal definition while the second is usually easier to calculate
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v

The first is the

2
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A
straight line has curvature 0

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