Chapter 1 Topics in Analytic Geometry

•••••••MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 74s = 3s = 2s = 1Reference points = −1s = −2s = −3CLet us now treat s as a variable. As the value of s changes, the corresponding pointP moves along C and the coordinate of P become functions of s. Thus, in 2-space thecoordinate of P are (x(s),y(s)), and in 3-space they are (x(s),y(s),z(s)). Therefore in2-space or 3-space the curve C is given by the parametric equationsx = x(s), y = y(s), or x = x(s), y = y(s), z = z(s)A parametric representation of a curve with arc length as the parameter is called an arclength parametrization of the curve.Example 4.19 Find the arc length parametrization of the circle x 2 + y 2 = a 2 with counterclockwiseorientation and (a,0) as the reference point.Solution .........Change of ParameterA change of parameter in a vector-valued function r(t) is a substitution t = g(τ) thatproduce a new vector-valued function r(g(τ)) have the same graph as r(t), but possiblytraced differently as the parameter τ increases.Example 4.20 Find a change of parameter t = g(τ) for the circlesuch thatr(t) = costi+sintj (0 ≤ t ≤ 2π)(a) the circle is traced counterclockwise as τ increases over the interval [0,1];(b) the circle is traced clockwise as τ increases over the interval [0,1].Solution .........Theorem 4.7 (Chain Rule) Let r(t) be a vector-valued function in 2-space or 3-spacethat is differentiable with respect to t. If t = g(τ) is a change of parameter in which g isdifferentiable with respect to τ, then r((g(τ)) is differentiable with respect to τ anddrdτ = dr dtdt dτ(4.26)