Chapter 1 Topics in Analytic Geometry

Chapter 4Vector-Valued Functions4.1 Introduction to Vector-Valued FunctionsParametric Curves in 3-SpaceRecall that if f and g are well-behaved functions, then the pair of parametric equationsx = f(t), y = g(t) (4.1)generatesacurve in2-spacethatistracedinaspecific directionastheparameter tincreases.We define this direction to be the orientation of the curve or the direction of increasingparameter, and we called the curve together with its orientation the graph of the parametricequations or the parametric curve represented by the equations. Analogously, if f, g and hare three well-behaved functions, then the parametric equationsx = f(t), y = g(t), z = h(t) (4.2)generatesacurve in3-spacethatistracedinaspecificdirectionastincreases. Asin2-space,this direction is called the orientation or direction of increasing parameter, and thecurve together with its orientation is called the graph of the parametric equations or theparametric curve represented by the equations. If no restrictions are state explicitlyor are implied by the equation, then it will be understood that t varies over the interval(−∞,+∞).Example 4.1 Describe the parametric curve represented by the equationsSolution .........x = 1−t, y = 3t, z = 2t.Example 4.2 Describe the parametric curve represented by the equationswhere a and c are positive constants.x = acosθ, y = asinθ, z = ct.61