calculusII-3

1、1Chapter 11Vector-Valued Functions (向量值函数向量值函数) In this Chapter, we will know some important concepts.Vector-Valued FunctionsThe Calculus of Vector-Valued FunctionsCurvature (曲率)Tangent and Normal Vectors2Example 1: Airplane Flight Path,Vectors Indicating Position3Section 11.1 Vector-Valued Function

2、s p Vector-Valued Functions (向量值函数)4Example 25Example 36Example 47Example 58Example 6: Matching9p Arc Length in 3R10Arc Length Formula11Example 7Intersection of Cone and Plane12Section 11.2 The Calculus of Vector-Valued Functionsp The Limits 13p The Continuity 1415p The Differentiation 1617p The Tan

3、gent Vectors 1819p The Integration 2021Section 11.4 Curvature (曲率)2223Same or Different? 24Notice: the rate of change of the unit tangent vectors with respect to arc length along the curve will give us a measure of sharpness.25p The Curvature (曲率)2627282930Section 11.5 Tangent and Normal Vectors (切向

4、量与法向量)3132p The Principal Unit Normal Vector (主要单位法向量) 33343536p The Binormal Vectors (副法线向量)3738p The Normal Plane and The Osculating Plane (法平面与密切平面) 39p The Osculating Circle (密切圆)4041p Tangential and Normal Components of Acceleration4243Section 11.6 Parametric Surfaces (参数曲面)l Motivation: Recall

5、 the circular helix given parametrically by 4445Example 146Example 2l Identify and sketch the surface defined by parametric equations47Hyperbolic Cosine and Sine Functionsl For parametric equations of hyperboloids and hyperbolic paraboloids, it is convenient to use the hyperbolic functions cosh x and sinh x given by48Example 3l Identify and sketch the surface defined by parametric equations49Example 4l Find a parametric representation of the hyperbolic paraboloid50Exam