Multivariable CalculusOliver KnillHarvard Summer School 2010AbstractThis is an extended syllabus for this summer. It tells the story of the entire coursein a condensed form. These 8 pages can be a guide through the semester. The materialis arranged in 6 chapters and delivered in the 6 weeks of the course. Each chapter has4 sections, two sections for each day. While it make sense to read in a text book besidefollowing the lectures, I want you to focus on the lectures. Textbooks have a lot ofadditional material and notation. Some of it can distract, other can even confuse. Whilea selected read in an other source can be helpful to get a second opinion and reconciliationof confusion and merging of different sources is an important aspect of learning, it can besufficient and save time, to focus on the lectures delivered in class. It goes without sayingthat homework is extremely important. Mathematics can only be learned by solvingproblems.Chapter 1. Geometry and SpaceSection 1.1: Space, distance, geometrical objectsAfter an overview over the syllabus, we use coordinates like P = (3, 4, 5) to describe pointsP in space. As promoted by Descartes in the 1 6’th century, g eometry can be describedalgebraically when a coordinate system is introduced. A fundamental notion is the distanced(P, Q) =p(x −a)2+ (y − b)2+ (z − c)2between two points P = (x, y, z) and Q = (a, b, c).This formula makes use of Pythagoras theorem. In order to g et a feel about space, we look atsome geometric objects in space. We will focus on simple examples like cylinders and spheresand learn how to find the center and radius of a sphere given as a quadratic expression inx, y, z. This method is called the completion of the square and is based on one of the oldesttechniques discovered in mathematics.Section 1.2: Vectors, dot product, projectionsTwo points P, Q in space define a vector~P Q at P . It has its head at Q and its tail at P . Thevector connects the initial point P with the end point Q. Vectors can be attached everywherein space, but they are identified if they have the same length and the same direction. Vectorscan describe velocities, forces or color or data. The components of a vector~P Q connectinga point P = (a, b, c) with a point Q = (x, y, z) are the entries of the vector hx −a, y −b, z −ci.1Examples of vectors are the zero vector~0 = h0, 0, 0i, and the standard basis vectors~i = h1 , 0, 0i,~j = h0, 1 , 0i,~k = h0, 0, 1i. Addition, subtract ion and scalar multiplicationof vectors can be done both geometrically and algebraically. The dot product ~v · ~w betweentwo vectors r esults is a scalar. It allows to define the length |~v| =√~v ·~v of a vector. Thetrigonometric cos-formula leads to the angle formula ~v · ~w = |~v||~w|cos α. By the Cauchy-Schwarz inequality we can define the angle between two vectors using this formula. Vectorssatisfying ~v · ~w = 0 are called perpendicular. Pythagoras formula |~v + ~w|2= |~v|2+ |~w|2follows now f r om the definitions.Section 1.3: The cross product and triple scalar productAfter a short review of the dot product we introduce the cross product ~v × ~w of two vectors~v = ha, b, ci and ~w = hp, q, ri in space. This new vector hbr−cq, cp−ar, aq −bpi is perpendicularto both vectors ha, b, ci and hp, q, ri. The product can be valuable for many things: it is usefulfor example to compute areas of parallelograms, the distance between a point and a line, orto construct a plane through three points or to intersect two planes. We prove a formula|~v × ~w| = |~v||~w|sin(α) for a quantity which is geometrically the area of the parallelepipedspanned by ~v and ~w. Finally, we look at the triple scalar product (~u × ~v) · ~w which is ascalar and is the signed volume of the par allelepiped spanned by ~u,~v and ~w. Its sign tellsabout the orientation of the coordinate system defined by the three vectors. The triple scalarproduct is zero if and only if the three vectors a r e in a common plane.Section 1.4: Lines, planes and distancesBecause the ha, b, ci = ~n = ~u × ~v is perpendicular to ~x − ~w if ~x, ~w are both in the planespanned by ~u and ~v, we are led to the equation ax + by + cz = d of the pla ne. Planes can bevisualized by their traces, the intersection with coordinate planes a s well as their intercepts,the intersection with coordinate axes. We often know the normal vector ~n = ha, b, ci to aplane and can determine the constant d by plugging in a known point (x, y, z) on equationax + by + cz = d. We introduce lines by the parameterization ~r(t) =~OP + t~v, where P is apoint on the line and ~v = ha, b, ci is a vector telling the direction of the line. If P = (o, p, q),then (x − o)/a = (y − p)/b = (z − q)/c is called the symmetric equation of a line. It canbe interpreted as the intersection of two planes. As an application of dot and cross products,we look at various distance formulas. Especially, we compute the distance from a point to aplane, the distance from a point to a line or the distance between two lines. We will also seehow to compute distances between points, lines, planes, cylinders and spheres.Chapter 2. Curves and SurfacesSection 2.1: Functions, level surfaces, quadricsWe first focus on functions f(x, y) of two variables. The graph of a function f (x, y) of twovariables is defined as the set of points (x, y, z) for which z − f(x, y) = 0. We look at a f ew2examples and match some graphs with functions f(x, y). Traces, the intersection of the gr aphwith the coordinate planes as well as generalized traces like f(x, y) = c which are called levelcurves of f help to visualize surfaces. The set of all level curves forms a so called contourmap. After a short review of conic sections like ellipses, parabola and hyperb ola in twodimensions, we look at more general surfaces of the form g(x, y, z) = 0. We start with knownexamples like the sphere and the plane. If g(x, y, z) is a function which only involves linearand quadratic terms, t he level surface is called a quadric. Important quadrics are spheres,ellipsoids, cones, paraboloids, cylinders as well as various hyperbo loids.Section 2.2: Parametric surfacesSurfaces can be described in two fundamental ways: implicitly or parametrically. Implicitdescriptions g(x, y, z) = 0 like x2+ y2+ z2− 1 = 0 have been introduced a lr eady earlier inthe course. We look now at parametrized surfaces ~r(u, v) = hx(u, v), y(u, v), z( u, v)i likethe sphere ~r(θ, φ) = hρ cos(θ) sin(φ), ρ sin(θ)
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