Dr.-Ing. Erwin Sitompul President University Lecture 1 Multivariable Calculus President UniversityErwin SitompulMVC 1/1

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

President UniversityErwin SitompulMVC 1/3 Grade Policy Multivariable Calculus Final Grade = 5% Homework + 30% Quizzes + 30% Midterm Exam + 40% Final Exam + Extra Points Homeworks will be given in fairly regular basis. The average of homework grades contributes 5% of final grade. Homeworks are to be submitted on A4 papers, otherwise they will not be graded. Homeworks must be submitted on time. If you submit late, < 10 min.  No penalty 10 – 60 min.  –20 points > 60 min.  –40 points There will be 3 quizzes. Only the best 2 will be counted. The average of quiz grades contributes 30% of the final grade. Midterm and final exam schedule will be announced in time. Make up of quizzes and exams will be held one week after the schedule of the respective quizzes and exams.

President UniversityErwin SitompulMVC 1/4 Grade Policy Multivariable Calculus 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). Extra points will be given if you solve a problem in front of the class. You will earn 1, 2, or 3 points. You are responsible to read and understand the lecture slides. I am responsible to answer your questions. Heading of Homework Papers (Required) Multivariable Calculus Homework 2 Ranran Agustin March No. 5. Answer:

President UniversityErwin SitompulMVC 1/5 Chapter 12 Vectors and the Geometry of Space

President UniversityErwin SitompulMVC 1/6 The Cartesian coordinate system To locate a point in space, we use three mutually perpendicular coordinate axes, arranged as in the figure below Three-Dimensional Coordinate SystemsChapter 12 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. Cartesian coordinates for space are also called rectangular coordinates.

President UniversityErwin SitompulMVC 1/7 The Cartesian coordinate system 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. The three planes meet at the origin (0,0,0). The origin is also identified by simply 0 or sometimes the letter O Three-Dimensional Coordinate SystemsChapter 12 The three coordinate planes x =0, y =0, and z =0 divide space into eight cells called octants.

President UniversityErwin SitompulMVC 1/8 The Cartesian coordinate system 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 x-axis. The y- and z-coordinates can be any numbers. The similar consideration can be made for planes perpendicular to the y-axis or z-axis Three-Dimensional Coordinate SystemsChapter 12 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. A point (x,y,z) lies on this line if and only if x = 2 and y = 3. The similar consideration can be made for other plane intersections.

President UniversityErwin SitompulMVC 1/9 The Cartesian coordinate system Example 12.1 Three-Dimensional Coordinate SystemsChapter 12

President UniversityErwin SitompulMVC 1/10 Distance and Spheres in Space 12.1 Three-Dimensional Coordinate SystemsChapter 12

President UniversityErwin SitompulMVC 1/11 Distance and Spheres in Space Example 12.1 Three-Dimensional Coordinate SystemsChapter 12

President UniversityErwin SitompulMVC 1/12 Distance and Spheres in Space 12.1 Three-Dimensional Coordinate SystemsChapter 12

President UniversityErwin SitompulMVC 1/13 Distance and Spheres in Space Example 12.1 Three-Dimensional Coordinate SystemsChapter 12

President UniversityErwin SitompulMVC 1/14 Component Form 12.2 VectorsChapter 12 A quantity such as force, displacement, or velocity is called a vector and is represented by a directed line segment. 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 UniversityErwin SitompulMVC 1/15 Component Form 12.2 VectorsChapter 12

President UniversityErwin SitompulMVC 1/16 Component Form 12.2 VectorsChapter 12

President UniversityErwin SitompulMVC 1/17 Component Form 12.2 VectorsChapter 12 Example

President UniversityErwin SitompulMVC 1/18 Component Form 12.2 VectorsChapter 12 Example

President UniversityErwin SitompulMVC 1/19 Vector Algebra Operations 12.2 VectorsChapter 12

President UniversityErwin SitompulMVC 1/20 Vector Algebra Operations 12.2 VectorsChapter 12 Example

President UniversityErwin SitompulMVC 1/21 Vector Algebra Operations 12.2 VectorsChapter 12

President UniversityErwin SitompulMVC 1/22 Vector Algebra Operations 12.2 VectorsChapter 12 Example

President UniversityErwin SitompulMVC 1/23 Unit Vectors 12.2 VectorsChapter 12 A vector v of length 1 is called a unit vector. The standard unit vectors are Any vector v  can be written as a linear combination of the standard unit vectors as follows: The unit vector in the direction of any vector v is called the direction of the vector, denoted as v/|v|.

President UniversityErwin SitompulMVC 1/24 Unit Vectors 12.2 VectorsChapter 12 Example

President UniversityErwin SitompulMVC 1/25 Unit Vectors 12.2 VectorsChapter 12 Example

President UniversityErwin SitompulMVC 1/26 Midpoint of a Line Segment 12.2 VectorsChapter 12

President UniversityErwin SitompulMVC 1/27 Angle Between Vectors 12.3 The Dot ProductChapter 12

President UniversityErwin SitompulMVC 1/28 Angle Between Vectors 12.3 The Dot ProductChapter 12 Example

President UniversityErwin SitompulMVC 1/29 Angle Between Vectors 12.3 The Dot ProductChapter 12 Example

President UniversityErwin SitompulMVC 1/30 Angle Between Vectors 12.3 The Dot ProductChapter 12 Example

President UniversityErwin SitompulMVC 1/31 Perpendicular (Orthogonal) Vectors 12.3 The Dot ProductChapter 12 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 Example

President UniversityErwin SitompulMVC 1/32 Dot Product Properties and Vector Projections 12.3 The Dot ProductChapter 12

President UniversityErwin SitompulMVC 1/33 Dot Product Properties and Vector Projections 12.3 The Dot ProductChapter 12

President UniversityErwin SitompulMVC 1/34 Dot Product Properties and Vector Projections 12.3 The Dot ProductChapter 12 Example

President UniversityErwin SitompulMVC 1/35 Work 12.3 The Dot ProductChapter 12

President UniversityErwin SitompulMVC 1/36 Writing a Vector as a Sum of Orthogonal Vectors 12.3 The Dot ProductChapter 12

President UniversityErwin SitompulMVC 1/37 Writing a Vector as a Sum of Orthogonal Vectors 12.3 The Dot ProductChapter 12 Example The force parallel to v The force orthogonal / perpendicular to v

President UniversityErwin SitompulMVC 1/38 Homework The Dot ProductChapter 12 Exercise 12.1, No. 37. Exercise 12.1, No. 50. Exercise 12.2, No. 24. Exercise 12.3, No. 2. Exercise 12.3, No. 22. Due: Next week, at