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Han_Multivariable calculus

# Han_Multivariable calculus - MULTIVARIABLE CALCULUS...

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MULTIVARIABLE CALCULUS XIAOLONG HAN Abstract. This lecture note is designed for a one-semester course in Calculus III: Mul- tivariable Calculus. We follow the textbook [S] closely and homework is assigned from it, too. I also use [A] as an excellent reference. Introduction This course covers vectors and multivariable calculus. Topics include vectors and their calculus in two- and three-dimensional spaces, parametric curves, partial derivatives and multiple integrals. There are two general interpretations: geometrical and analytical, the connection between them lies in every subject in this course, so do not hesitate to exploit your geometric visualization and analytic computation! As its name suggests, multivariable calculus is the extension of calculus to more than one variables. That is, in single variable calculus we study functions of a single independent variable y = f ( x ) . In multivariable calculus we study functions of two or more independent variables, e.g. z = f ( x, y ) or w = f ( x, y, z ) . These functions are interesting in their own right, but they are also essential for de- scribing the physical world. We are also interested in vector functions, i.e., the functions with vector values as outputs. By the end of the course we will know how to differentiate and integrate functions of several variables. In single variable calculus the Fundamental Theorem of Calculus relates derivatives to integrals. We will see something similar in multivariable calculus and the capstone to the course will be the three theorems (Green’s, Stokes’ and Gauss’) that do this. Contents Introduction 1 1. Vectors and the geometry of space 2 1.1. Three-dimensional rectangular coordinate systems and distance formula 2 1.2. Vectors and their basic properties 4 1.3. The dot product 6 1.4. The cross product 8 1.5. Lines and planes in three-dimensional space 10 2. Vector functions 13 1

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2 XIAOLONG HAN 2.1. Vector functions and space curves 13 2.2. Derivatives and integrals of vector functions 14 2.3. Arc length 16 2.4. Motion in space: velocity and acceleration 16 3. Partial derivatives 17 3.1. Functions of several variables 17 3.2. Partial derivatives 19 3.3. Tangent planes and linear approximations 20 3.4. The chain rule and implicit function theorem 21 3.5. Directional derivatives and the gradient vector 22 3.6. Maximum and minimum values 23 4. Multiple integrals 25 4.1. Double integrals over rectangles 25 4.2. Iterated integrals 26 4.3. Double integrals over general regions 27 4.4. Double integrals in polar coordinates 28 4.5. Triple integrals over rectangular boxes 29 4.6. Change of variables in double integrals 29 References 30 1. Vectors and the geometry of space Recall xy -plane, two-dimensional space, as the frame of single variable calculus, we introduce three-dimensional space as the frame of multivariable calculus (strictly, two variables). The frame includes coordinate systems, vectors and their calculus. Objectives in this chapter are (1) Building three-dimensional rectangular coordinate systems and deriving distance formula; (2) Defining vectors in two- and three-dimensional spaces and introducing vector cal-
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