Multivariable Calculus Lecture 1 Dr Ing Erwin Sitompul

Multivariable Calculus Lecture 1 Dr. -Ing. Erwin Sitompul President University http: //zitompul. wordpress. com President University Erwin Sitompul MVC 1/1

Multivariable Calculus Textbook and Syllabus Textbook: “Thomas’ Calculus”, 11 th Edition, George B. Thomas, Jr. , et. al. , Pearson, 2005. Syllabus: n Chapter 12: n Chapter 13: n Chapter 14: n Chapter 15: n Chapter 16: Vectors and the Geometry of Space Vector-Valued Functions and Motion in Space Partial Derivatives Multiple Integrals Integration in Vector Fields President University Erwin Sitompul MVC 1/2

Multivariable Calculus Grade Policy Final Grade = 5% Homework + 30% Quizzes + 30% Midterm Exam + 40% Final Exam + Extra Points n Homeworks will be given in fairly regular basis. The average of homework grades contributes 5% of final grade. n Homeworks are to be submitted on A 4 papers, otherwise they will not be graded. n Homeworks must be submitted on time. If you submit late, < 10 min. No penalty 10 – 60 min. – 20 points > 60 min. – 40 points n There will be 3 quizzes. Only the best 2 will be counted. The average of quiz grades contributes 30% of the final grade. n Midterm and final exam schedule will be announced in time. n Make up of quizzes and exams will be held one week after the schedule of the respective quizzes and exams. President University Erwin Sitompul MVC 1/3

Multivariable Calculus Grade Policy n The score of a make up quiz or exam, upon discretion, can be multiplied by 0. 9 (i. e. , the maximum score for a make up is then 90). n Extra points will be given if you solve a problem in front of the class. You will earn 1, 2, or 3 points. n You are responsible to read and understand the lecture slides. I am responsible to answer your questions. Multivariable Calculus Homework 2 Ranran Agustin 009200700008 21 March 2009 13. 1 No. 5. Answer: . . . . n Heading of Homework Papers (Required) President University Erwin Sitompul MVC 1/4

Chapter 12 Vectors and the Geometry of Space President University Erwin Sitompul MVC 1/5

Chapter 12 12. 1 Three-Dimensional Coordinate Systems The Cartesian coordinate system n To locate a point in space, we use three mutually perpendicular coordinate axes, arranged as in the figure below. n The Cartesian coordinates (x, y, z) of a point P in space are the number at which the planes through P perpendicular to the axes cut the axes. n Cartesian coordinates for space are also called rectangular coordinates. President University Erwin Sitompul MVC 1/6

Chapter 12 12. 1 Three-Dimensional Coordinate Systems The Cartesian coordinate system n The planes determined by the coordinates axes are the xy-plane, where z = 0; the yz-plane, where x = 0; and the xz-plane, where y = 0. n The three planes meet at the origin (0, 0, 0). n The origin is also identified by simply 0 or sometimes the letter O. n The three coordinate planes x = 0, y = 0, and z = 0 divide space into eight cells called octants. President University Erwin Sitompul MVC 1/7

Chapter 12 12. 1 Three-Dimensional Coordinate Systems The Cartesian coordinate system n The points in a plane perpendicular to the x-axis all have the same x-coordinate, which is the number at which that plane cuts the xaxis. The y- and z-coordinates can be any numbers. n The similar consideration can be made for planes perpendicular to the y-axis or z-axis. n The planes x = 2 and y = 3 on the next figure intersect in a line parallel to the z-axis. This line is described by a pair of equations x = 2, y = 3. n A point (x, y, z) lies on this line if and only if x = 2 and y = 3. n The similar consideration can be made for other plane intersections. President University Erwin Sitompul MVC 1/8

Chapter 12 12. 1 Three-Dimensional Coordinate Systems The Cartesian coordinate system n Example President University Erwin Sitompul MVC 1/9

Chapter 12 12. 1 Three-Dimensional Coordinate Systems Distance and Spheres in Space President University Erwin Sitompul MVC 1/10

Chapter 12 12. 1 Three-Dimensional Coordinate Systems Distance and Spheres in Space n Example President University Erwin Sitompul MVC 1/11

Chapter 12 12. 1 Three-Dimensional Coordinate Systems Distance and Spheres in Space President University Erwin Sitompul MVC 1/12

Chapter 12 12. 1 Three-Dimensional Coordinate Systems Distance and Spheres in Space n Example President University Erwin Sitompul MVC 1/13

Chapter 12 12. 2 Vectors Component Form n A quantity such as force, displacement, or velocity is called a vector and is represented by a directed line segment. n The arrow points in the direction of the action and its length gives the magnitude of the action in terms of a suitable chosen unit. President University Erwin Sitompul MVC 1/14

Chapter 12 12. 2 Vectors Component Form President University Erwin Sitompul MVC 1/15

Chapter 12 12. 2 Vectors Component Form President University Erwin Sitompul MVC 1/16

Chapter 12 12. 2 Vectors Component Form n Example President University Erwin Sitompul MVC 1/17

Chapter 12 12. 2 Vectors Component Form n Example President University Erwin Sitompul MVC 1/18

Chapter 12 12. 2 Vectors Vector Algebra Operations President University Erwin Sitompul MVC 1/19

Chapter 12 12. 2 Vectors Vector Algebra Operations n Example President University Erwin Sitompul MVC 1/20

Chapter 12 12. 2 Vectors Vector Algebra Operations President University Erwin Sitompul MVC 1/21

Chapter 12 12. 2 Vectors Vector Algebra Operations n Example President University Erwin Sitompul MVC 1/22

Chapter 12 12. 2 Vectors Unit Vectors n A vector v of length 1 is called a unit vector. The standard unit vectors are n Any vector v = <v 1, v 2, v 3> can be written as a linear combination of the standard unit vectors as follows: n The unit vector in the direction of any vector v is called the direction of the vector, denoted as v/|v|. President University Erwin Sitompul MVC 1/23

Chapter 12 12. 2 Vectors Unit Vectors n Example President University Erwin Sitompul MVC 1/24

Chapter 12 12. 2 Vectors Unit Vectors n Example President University Erwin Sitompul MVC 1/25

Chapter 12 12. 2 Vectors Midpoint of a Line Segment President University Erwin Sitompul MVC 1/26

Chapter 12 12. 3 The Dot Product Angle Between Vectors President University Erwin Sitompul MVC 1/27

Chapter 12 12. 3 The Dot Product Angle Between Vectors n Example President University Erwin Sitompul MVC 1/28

Chapter 12 12. 3 The Dot Product Angle Between Vectors n Example President University Erwin Sitompul MVC 1/29

Chapter 12 12. 3 The Dot Product Angle Between Vectors n Example President University Erwin Sitompul MVC 1/30

Chapter 12 12. 3 The Dot Product Perpendicular (Orthogonal) Vectors n Two nonzero vectors u and v are perpendicular or orthogonal if the angle between them is π/2. For such vectors, we have u · v =|u||v|cosθ = 0 n Example President University Erwin Sitompul MVC 1/31

Chapter 12 12. 3 The Dot Product Properties and Vector Projections President University Erwin Sitompul MVC 1/32

Chapter 12 12. 3 The Dot Product Properties and Vector Projections President University Erwin Sitompul MVC 1/33

Chapter 12 12. 3 The Dot Product Properties and Vector Projections n Example President University Erwin Sitompul MVC 1/34

Chapter 12 12. 3 The Dot Product Work President University Erwin Sitompul MVC 1/35

Chapter 12 12. 3 The Dot Product Writing a Vector as a Sum of Orthogonal Vectors President University Erwin Sitompul MVC 1/36

Chapter 12 12. 3 The Dot Product Writing a Vector as a Sum of Orthogonal Vectors n Example The force parallel to v President University The force orthogonal / perpendicular to v Erwin Sitompul MVC 1/37

Chapter 12 12. 3 The Dot Product Homework 1 n Exercise n Exercise 12. 1, 12. 2, 12. 3, No. No. No. 37. 50. 24. 2. 22. n Due: Next week, at 17. 15. President University Erwin Sitompul MVC 1/38