Chapter 1 Topics in Analytic Geometry

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••MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 74A change of parameter t = g(τ) in which r(g(τ)) is smooth is called a smooth changeof parameter. Smooth change of parameter fall into two categories—those for whichdt/dτ > 0forallτ (calledpositive changes of parameter)andthoseforwhichdt/dτ < 0for all τ (called negative changes of parameter). A positive changes of parameterpreserves theorientationofaparametriccurve, andanegativechangesofparameterreversesit.Finding Arc Length ParametrizationsTheorem 4.8 Let C be the graph of a smooth vector-valued function r(t) in 2-space or 3-space, and let r 0 (t) be any point on C. Then the following formula defines a positive changeof parameter from t to s, where s is an arc length parameter having r 0 (t) as its referencepoint:∫ t∥ ∥∥∥ drs =t 0du∥ du (4.27)yr(t 0 )sr(t)CxWhen needed, Formula (4.27) can be expressed in component form ass =s =∫ t√ (dx ) 2+t 0du∫ t√ (dx ) 2+t 0du( ) 2 dy+du( ) 2 dydu 2-space (4.28)du( ) 2 dzdu 3-space (4.29)duExample 4.21 Find the arc length parametrization of the circular helixr = costi+sintj+tkthat has reference point r(0) = (1,0,0) and has the same orientation as the given helix.Solution .........Example 4.22 A bug walks along the trunk of a tree following a path modeled by thecircular helix in Example 4.21. The bug starts at the reference point (1,0,0) and walks upthe helix for a distance of 10 units. What are the bug’s final coordinates?
MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 75Solution .........Example 4.23 Find the arc length parametrization of the linex = 2t+1, y = 3t−2that has the same orientation as the given line and uses (1,−2) as the reference point.Solution .........Properties of Arc Length ParametrizationsTheorem 4.9(a) If C is the graph of a smooth vector-valued function r(t) in 2-space or 3-space, wheret is a general parameter, and if s is the arc length parameter for C defined by Formula(4.27), then for every value of t the tangent vector has lengthdr∥dt∥ = ds(4.30)dt(b) If C is the graph of a smooth vector-valued function r(t) in 2-space or 3-space, wheres is an arc length parameter, then for every value of s the tangent vector to C haslengthdr∥ds∥ = 1 (4.31)(c) If C is the graph of a smooth vector-valued function r(t) in 2-space or 3-space, andif ‖dr/dt‖ = 1 for every value of t, then for any value of t 0 in the domain of r, theparameter s = t−t 0 is an arc length parameter that has its reference point at the pointon C where t = t 0 .The component forms of Formulas (4.30) and (4.31) are as follow:∥ √ (dx )ds ∥∥∥dt = dr2 dt∥ = +dt∥ √ (dx )ds ∥∥∥dt = dr2 dt∥ = +dt√ (dx ) dr2 ∥ds∥ = +ds√ (dx ) dr2 ∥ds∥ = +ds( ) 2 dy+ds( ) 2 dy+dt( ) 2 dy2-space (4.32)dt( ) 2 dz3-space (4.33)dt( ) 2 dy= 1 2-space (4.34)ds( ) 2 dz= 1 3-space (4.35)ds
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Chapter 1 Topics in Analytic Geometry

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