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1. Multivariable Calculus Textbook and Syllabus Textbook: “Thomas’ Calculus”, 11thEdition, George B. Thomas, Jr., et. al., Pearson, 2005. • 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

2. Multivariable Calculus Grade Policy • 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.

3. Multivariable Calculus Grade Policy • 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. Multivariable CalculusHomework 2Ranran Agustin00920070000821 March 200913.1 No. 5. Answer: . . . . . . . . • Heading of Homework Papers (Required)

4. Chapter 12 Vectors and the Geometry of Space

5. Chapter 12 12.1 Three-Dimensional Coordinate Systems The Cartesian coordinate system • To locate a point in space, we use three mutually perpendicular coordinate axes, arranged as in the figure below. • 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.

6. Chapter 12 12.1 Three-Dimensional Coordinate Systems 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. • The three coordinate planes x=0, y =0, and z =0 divide space into eight cells called octants.

7. Chapter 12 12.1 Three-Dimensional Coordinate Systems 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. • 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.

8. Chapter 12 12.1 Three-Dimensional Coordinate Systems The Cartesian coordinate system • Example

9. Chapter 12 12.1 Three-Dimensional Coordinate Systems Distance and Spheres in Space

10. Chapter 12 12.1 Three-Dimensional Coordinate Systems Distance and Spheres in Space • Example

11. Chapter 12 12.1 Three-Dimensional Coordinate Systems Distance and Spheres in Space

12. Chapter 12 12.1 Three-Dimensional Coordinate Systems Distance and Spheres in Space • Example

13. Chapter 12 12.2 Vectors Component Form • 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.

14. Chapter 12 12.2 Vectors Component Form

15. Chapter 12 12.2 Vectors Component Form

16. Chapter 12 12.2 Vectors Component Form • Example

17. Chapter 12 12.2 Vectors Component Form • Example

18. Chapter 12 12.2 Vectors Vector Algebra Operations

19. Chapter 12 12.2 Vectors Vector Algebra Operations • Example

20. Chapter 12 12.2 Vectors Vector Algebra Operations

21. Chapter 12 12.2 Vectors Vector Algebra Operations • Example

22. Chapter 12 12.2 Vectors Unit Vectors • A vector v of length 1 is called a unit vector. The standard unit vectors are • Any vector v= <v1,v2,v3> 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|.

23. Chapter 12 12.2 Vectors Unit Vectors • Example

24. Chapter 12 12.2 Vectors Unit Vectors • Example

25. Chapter 12 12.2 Vectors Midpoint of a Line Segment

26. Chapter 12 12.3 The Dot Product Angle Between Vectors

27. Chapter 12 12.3 The Dot Product Angle Between Vectors • Example

28. Chapter 12 12.3 The Dot Product Angle Between Vectors • Example

29. Chapter 12 12.3 The Dot Product Angle Between Vectors • Example

30. Chapter 12 12.3 The Dot Product Perpendicular (Orthogonal) Vectors • 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

31. Chapter 12 12.3 The Dot Product Dot Product Properties and Vector Projections

32. Chapter 12 12.3 The Dot Product Dot Product Properties and Vector Projections

33. Chapter 12 12.3 The Dot Product Dot Product Properties and Vector Projections • Example

34. Chapter 12 12.3 The Dot Product Work

35. Chapter 12 12.3 The Dot Product Writing a Vector as a Sum of Orthogonal Vectors

36. Chapter 12 12.3 The Dot Product Writing a Vector as a Sum of Orthogonal Vectors • Example The force orthogonal / perpendicular to v The force parallel to v

37. Chapter 12 12.3 The Dot Product Homework 1 • 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 17.15.