# length - ARC LENGTH O Knill Math21a HOMEWORK Section 12.3...

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Unformatted text preview: ARC LENGTH O. Knill, Math21a HOMEWORK: Section 12.3: 12,14,42, 56, 58 PLANE CURVE vector r ( t ) = ( x ( t ) , y ( t ) ) position vector r ′ ( t ) = ( x ′ ( t ) , y ′ ( t ) ) velocity | vector r ′ ( t ) | = | ( x ′ ( t ) , y ′ ( t )) | speed vector r ′′ ( t ) = ( x ′′ ( t ) , y ′′ ( t ) ) acceleration vector r ′′′ ( t ) = ( x ′′′ ( t ) , y ′′′ ( t ) ) jerk SPACE CURVE vector r ( t ) = ( x ( t ) , y ( t ) , z ( t ) ) position vector r ′ ( t ) = ( x ′ ( t ) , y ′ ( t ) , z ′ ( t ) ) velocity | vector r ′ ( t ) | = | ( x ′ ( t ) , y ′ ( t )) | speed vector r ′′ ( t ) = ( x ′′ ( t ) , y ′′ ( t ) , z ′′ ( t ) ) acceleration vector r ′′′ ( t ) = ( x ′′′ ( t ) , y ′′′ ( t ) , z ′′′ ( t ) ) jerk TANGENT DIRECTION: The velocity vector vector r ′ ( t ) is tangent to the curve at vector r ( t ) because vector r ′ ( t ) ∼ vector r ( t + h ) − vector r ( t ) h . Note that the second derivative vector r ′′ ( t ) can point in any direction. It is by Newtons law proportional to the force acting on the body. ARC LENGTH. If t ∈ [ a, b ] mapsto→ vector r ( t ) with velocity vector r ′ ( t ) and speed | vector r ′ ( t ) | , then integraltext b a | vector r ′ ( t ) | dt is called the arc length of the curve . For space curves for example, this can be written out as L = integraltext b a radicalbig x ′ ( t ) 2 + y ′ ( t ) 2 + z ′ ( t ) 2 dt Note that the arc length is a scalar. The integralNote that the arc length is a scalar....
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