This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Chapter 12: VectorValued Functions Summary: Having defined vectors and their properties, now vectorvalued functions are considered. These are simply functions that have two or more out puts that are the components of a vector. As functions, the usual ideas of limits, derivatives, and integration are applied to vectorvalued functions. One natural way to describe a vectorvalued function is as a parameterization where each com ponent is a function of a parameter such as time, t . Parameterization and derivatives together lead to the ideas of unit tangent and unit normal vectors and ways to describe the curvature of a given curve that can be defined parametrically. The chapter finishes by discussing motion along a curve and how this relates to Kepler’s Laws of Planetary Motion. OBJECTIVES: After reading and working through this chapter you should be able to do the following: 1. Identify the curve of a parametric equation (§12 . 1). 2. Represent a curve as a vectorvalued function (§12 . 1). 3. Evaluate limits, derivatives, and integrals of vectorvalued functions (§12 . 2). 4. Take derivatives of dot and cross products (§12 . 2). 5. Find an arc length parameterization of a curve (§12 . 3). 6. Use the chain rule to take the derivative of a vectorvalued function (§12 . 3). 7. Find unit normal, unit tangent, and unit binormal vectors of a curve or vectorvalued function in 2space or 3space (§12 . 4). 8. Calculate the curvature of a parametric curve or vectorvalued function at a point (§12 . 5). 9. Relate position, velocity, and acceleration vectors of a particle traveling along a curve (§12 . 6). 10. Find the normal and tangential components of acceleration (§12 . 6). 11. Understand the relationship between acceleration, velocity, and curvature (§12 . 5 , 12 . 6). 12. Understand and apply Kepler’s three laws of planetary motion (§12 . 7). 255 256 12.1 Introduction to VectorValued Functions PURPOSE: To describe vectorvalued functions which are them selves vectors and to introduce how vectorvalued functions may be defined parametrically. A function that has one input and only one output is called a scalarvalued func scalarvalued function tion since the output is a scalar value. This section describes how curves can be describe parametrically and how curves may represent functions that have vector values or vectorvalued functions . vectorvalued function In the last chapter, lines in 3space were described using a parameterization for each coordinate in terms of some parameter or variable such as t . Curves in 3 curves in 3space space can also be parameterized in the same way. Each coordinate of a point on the curve is determined by some parameter such as t ....
View Full
Document
 Spring '10
 lee

Click to edit the document details