Unit 2 Vector Calculus In this unit, we consider vector functions, differentiation formulas, curves, arc length, curvilinear motion, vector calculus in mechanics, planetary motion and curvature. Note: Unit 2 is based on Chapter 13 of the textbook, Salas and Hille s Calculus: Several Variables, 7th ed., revised by Garret J. Etgen (New York: Wiley, 1995). All assigned readings and exercises are from that textbook, unless otherwise indicated. Objectives Detailed objectives are given in each of the sections listed below. 1 Vector Functions 2 Differentiation Formulas 3 Curves 4 Arc Length 5 Curvilinear Motion and Vector Calculus in Mechanics 6 Planetary Motion 7 Curvature Objective 1 a. differentiate a vector function f ( t). b. calculate the second derivative and the integral of a vector function f ( t). c. find the limit (if it exists) of a vector function f ( t) as t 0. d. sketch the curve traced by a vector function and identify the vector function that traces a given curve. e. identify the vector function f ( t), given its derivative f ( t). Mathematics 365 / Study Guide 9
Read Section 13.1, pages 839-846. Complete problems 1, 3, 5, 7, 11, 15, 19, 21, 23, 29, 37, 39, 43 45, 49, 51, 53 and 57, on pages 846-848. vector-valued function vector function components of f domain of f plane curve space curve limit of a vector function pinching theorem (see p. 656) limit rules differentiable continuous derivative of a vector function vector difference quotient integral of an vector function properties of the integral Before you proceed to Objective 2, make certain that you can meet each of the sub-objectives listed under Objective 1. 10 Calculus Several Variables
Objective 2 a. apply operations of vector algebra to vector functions with a common domain to form new vectors b. form scalar products and compositions of scalar functions and vector functions with a common domain c. determine whether combinations of vector and scalar functions are differentiable, and if so, find their derivatives. Read Section 13.2, pages 848-853. Complete all of the odd-numbered problems on pages 852-853. rules of derivatives of differentiable combinations of vector functions and scalar and vector functions prime notation and Leibniz notation [Note: In prime notation, the derivative of f is denoted by f, the second derivative by f, etc. ] component-by-component derivation component-free derivation definition of the derivative of a scalar function Before you proceed to Objective 3, make certain that you can meet each of the sub-objectives listed under Objective 2. Mathematics 365 / Study Guide 11
Objective 3 After completing this unit, you should be able to a. define a vector tangent to a given curve at a given point, and parameterize the corresponding tangent line. b. determine the angle between two intersecting curves. c. find equations for the unit tangent, the principle normal and the osculating plane for a given curve. d. reverse the sense of a curve. e. determine whether a given change of parameter is a sense-preserving or sense-reversing change. Read Section 13.3, pages 853-861. Complete problems 1, 5, 9, 11, 13, 15, 17, 21, 25, 27, 29, 31, 33, 35, 37, 39 and 43 on pages 862-863. differentiable curve oriented curve difference quotient direction vector for the tangent line tangent vector parameterization of the tangent line intersecting curves unit tangent vector principal normal vector osculating plane reverse the sense of a curve 12 Calculus Several Variables
sense-preserving change of parameter sense-reversing change of parameter Before you proceed to Objective 4, make certain that you can meet each of the sub-objectives listed under Objective 3. Objective 4 a. determine the length of a given curve. b. demonstrate various conclusions relating to arc length. Read Section 13.4, pages 862-868 and pages 869-870. Complete all of the odd-numbered problems 1-25 on pages 868-869. arc length arc length formula additivity of arc length Before you proceed to Objective 5, make certain that you can meet each of the sub-objectives listed under Objective 4. Mathematics 365 / Study Guide 13
Objective 5 After completing this unit, you should be able to a. define curvilinear motion in terms of vectors. b. solve problems relating to velocity, speed, acceleration, magnitude of the acceleration and angular momentum, in uniform circular motion about the origin, and in motion along a straight line. c. solve initial value problems involving free particles, charged particles, and objects having mass. d. demonstrate various conclusions relating to force, velocity, acceleration, angular momentum and torque. Read Section 13.5, pages 870-880. Complete problems 1, 3, 5, 7, 11, 15, 17, 18, 19, 20, 21, 23, 25 and 27 on pages 880-882. velocity acceleration speed (rate of change of arc distance with respect to time) angular velocity angular speed uniform circular motion Newton s second law of motion (vector form) conserved quantity conservation law momentum angular momentum torque 14 Calculus Several Variables
central force (radial force) initial value problem free particle Before you proceed to Objective 6, make certain that you can meet each of the sub-objectives listed under Objective 5. Objective 6 a. calculate the total force acting on an extended, three-dimensional object. b. derive Kepler s laws of planetary motion from Newton s laws of motion and gravitation. c. solve problems involving Kepler s laws of planetary motion. d. demonstrate various conclusions relating to planetary motion. Read Section 13.6, pages 882-887. Complete problems 1-5 on page 887. Kepler s laws of planetary motion centre of mass total external force Before you proceed to Objective 7, make certain that you can meet each of the sub-objectives listed under Objective 6. Mathematics 365 / Study Guide 15
Objective 7 a. find the curvature of a plane curve. b. find the radius of curvature of a plane curve at a given point. c. determine points of maximal and minimal curvature. d. find the curvature of a space curve. e. prove the Frenet formulas f. demonstrate various conclusions relating to curvature. Read Section 13.7, pages 888-894. Complete problems 1, 3, 7, 11, 15, 17, 19, 21, 25, 31, 33, 35, 39, 40, 47 and 48 on pages 894-896. Before you complete the first tutor-marked assignment, make certain that you can meet Objective 7. Assignment 1 Complete Tutor-marked Assignment 1, which you will find in the Assignments for Credit section of your Student Manual. Submit the completed assignment to your tutor for grading. Remember to include a Tutor-marked Exercise form, from the course package, with the assignment you submit, and to keep a copy of your work, at least a rough draft, in case the original is lost in the mail. 16 Calculus Several Variables