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## Textbook and Syllabus

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**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**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.**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)**Chapter 12**Vectors and the Geometry of Space**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.**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.**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.**Chapter 12**12.1 Three-Dimensional Coordinate Systems The Cartesian coordinate system • Example**Chapter 12**12.1 Three-Dimensional Coordinate Systems Distance and Spheres in Space**Chapter 12**12.1 Three-Dimensional Coordinate Systems Distance and Spheres in Space • Example**Chapter 12**12.1 Three-Dimensional Coordinate Systems Distance and Spheres in Space**Chapter 12**12.1 Three-Dimensional Coordinate Systems Distance and Spheres in Space • Example**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.**Chapter 12**12.2 Vectors Component Form**Chapter 12**12.2 Vectors Component Form**Chapter 12**12.2 Vectors Component Form • Example**Chapter 12**12.2 Vectors Component Form • Example**Chapter 12**12.2 Vectors Vector Algebra Operations**Chapter 12**12.2 Vectors Vector Algebra Operations • Example**Chapter 12**12.2 Vectors Vector Algebra Operations**Chapter 12**12.2 Vectors Vector Algebra Operations • Example**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|.**Chapter 12**12.2 Vectors Unit Vectors • Example**Chapter 12**12.2 Vectors Unit Vectors • Example**Chapter 12**12.2 Vectors Midpoint of a Line Segment**Chapter 12**12.3 The Dot Product Angle Between Vectors**Chapter 12**12.3 The Dot Product Angle Between Vectors • Example**Chapter 12**12.3 The Dot Product Angle Between Vectors • Example**Chapter 12**12.3 The Dot Product Angle Between Vectors • Example**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**Chapter 12**12.3 The Dot Product Dot Product Properties and Vector Projections**Chapter 12**12.3 The Dot Product Dot Product Properties and Vector Projections**Chapter 12**12.3 The Dot Product Dot Product Properties and Vector Projections • Example**Chapter 12**12.3 The Dot Product Work**Chapter 12**12.3 The Dot Product Writing a Vector as a Sum of Orthogonal Vectors**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**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.