curvature - CURVATURE Math21a HOMEWORK Section 12.4 10...

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Unformatted text preview: CURVATURE Math21a HOMEWORK: Section 12.4: 10, 46, (and on a different topic 13.1: 38, 46, 64) CURVATURE. If vector T ( t ) = vector r ′ ( t ) | / | vector r ′ ( t ) | unit tangent vector , define the curvature at the point vector r ( t ) as κ ( t ) = | vector T ′ ( t ) | | vector r ′ ( t ) | . CURVATURE FORMULA. The following formula for the curvature sheds more light on the curvature and is often easier to compute: κ ( t ) = | vector r ′ ( t ) × vector r ′′ ( t ) | | vector r ′ ( t ) | 3 The same formula holds for curves in the plane if we define the cross product in the plane as ( a, b ) × ( c, d ) = ad- bc . EXAMPLE. CIRCLE vector r ( t ) = ( r cos( t ) , r sin( t )). vector r ′ ( t ) = (- r sin( t ) , r cos( t )). | vector r ′ ( t ) | = r . vector T ( t ) = (- sin( t ) , cos( t )). vector r ′′ ( t ) = (- r cos( t ) ,- r sin( t )). vector T ′ ( t ) = (- cos( t ) ,- sin( t )). κ ( t ) = | vector T ′ ( t ) | / | vector r ′ ( t ) | = 1 /r ....
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This note was uploaded on 03/29/2008 for the course MATH 251 taught by Professor Skrypka during the Spring '08 term at Texas A&M.

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