2011_14f_Note (1) - MATH2011 Introduction to Multivariable...

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MATH2011Introduction to Multivariable CalculusDepartment of MathematicsThe Hong Kong University of Science and Technology2014 Fall Semester
Contents1Vectors and Vector-Valued Functions31.1Rectangular Coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31.2Vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31.3Lines in the Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61.4Planes in the Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71.5Curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .81.6Arc-Length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .111.7Quadric Surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .111.8Cylindrical and Spherical Coordinate Systems. . . . . . . . . . . . . . . . . . . . . . . . .112Functions of Several Variables152.1Functions of Two or More Variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .152.2Partial Derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .172.3The Chain Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .212.4Directional Derivatives and Gradients. . . . . . . . . . . . . . . . . . . . . . . . . . . . .232.5Tangent Planes and Normal Vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .242.6Extremums of Functions of Two Variables. . . . . . . . . . . . . . . . . . . . . . . . . . .253Multiple Integration273.1Double Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .273.2Triple Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .324Vector Calculus354.1Vector Fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .354.2Line Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .374.3Green’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .394.4Surface Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .414.5The Divergence Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .434.6Stoke’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .441
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Chapter 1Vectors and Vector-Valued Functions1.1Rectangular CoordinatesPoints in the space are presented by theircoordinates, similar to the case for points in a plane.Take three lines in the space which are mutually perpendicular and we call them thex-axis,y-axisand thez-axis. The intersection of these three lines is called the originO.Thexy-plane is the plane containing both thex-axis and they-axis, and its similar for theyz-planeand thexz-plane.The first (x), second (y), third (z) coordinates of a pointPare the perpendicular distances fromPto theyz-plane,xz-plane,xy-plane respectively.For a function in one variablef, the graph offis the set of points (x, y) in the plane satisfyingy=f(x). The graph is a geometric object closely related to the functionf.In general, iffis a function in two variables. The graph offis the set of points (x, y, z) in thespace satisfyingz=f(x, y). It is a surface in the space and is closely related to the study off.If (a, b, c) is a point in the space. The distance from the origin to this point isa2+b2+c2. Thisresult is done by applying the Pythagoras theorem for twice. More generally, the distance betweenthe points (a, b, c) and (α, β, γ) in the space is(a-α)2+ (b-β)2+ (c-γ)2.Example 1.1.1.The set of points (x, y, z) in the space satisfyingx2+y2+z2= 4 is the set of pointshaving a distance 2 from the origin. Thus, it is a sphere with center the origin and radius 2.Example 1.1.2.The set of points (x, y, z) in the space satisfyingy2+z2= 4 is the set of points havinga distance 2 from thex-axis. Thus, it is a cylinder with its axis thex-axis and radius 2.1.2VectorsDefinition 1.2.1.A vector is an ordered collection of numbers.There are some operations defined for vectors:Definition 1.2.2.Addition:(x1, ..., xn) + (y1, ..., yn) = (x1+y1, ..., xn+yn)Definition 1.2.3.Scalar multiplication:a(x1, ..., xn) = (ax1, ...axn)Theorem 1.2.1.For all vectorsx,y,zand for all numbersa,b,3
4(a)(b)Figure 1.1: (a) Example 1.1.1 and (b) Example 1.1.2.

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