Multivariable Calculus

Abstract

This is an extended syllabus for this summer. It tells the story of the entire course in a condensed form. These 8 pages can be a guide through the semester. The material is arranged in 6 chapters and delivered in the 6 weeks of the course. Each chapter has 4 sections, two sections for each day. While it make sense to read in a text book beside following the lectures, I want you to focus on the lectures. Textbooks have a lot of additional material and notation. Some of it can distract, other can even confuse. While a selected read in an other source can be helpful to get a second opinion and reconciliation of confusion and merging of different sources is an important aspect of learning, it can be sufficient and save time, to focus on the lectures delivered in class. It goes without saying that homework is extremely important. Mathematics can only be learned by solving problems. Chapter 1. Geometry and Space Section 1.1: Space, distance, geometrical objects After an overview over the syllabus, we use coordinates like P = (3, 4, 5) to describe points P in space. As promoted by Descartes in the 16’th century, geometry can be described algebraically when a coordinate system is introduced. A fundamental notion is the distance d(P,Q) = √ (x− a) + (y − b) + (z − c) between two points P = (x, y, z) and Q = (a, b, c). This formula makes use of Pythagoras theorem. In order to get a feel about space, we look at some geometric objects in space. We will focus on simple examples like cylinders and spheres and learn how to find the center and radius of a sphere given as a quadratic expression in x, y, z. This method is called the completion of the square and is based on one of the oldest techniques discovered in mathematics. Section 1.2: Vectors, dot product, projections Two points P,Q in space define a vector ~ PQ at P . It has its head at Q and its tail at P . The vector connects the initial point P with the end point Q. Vectors can be attached everywhere in space, but they are identified if they have the same length and the same direction. Vectors can describe velocities, forces or color or data. The components of a vector ~ PQ connecting a point P = (a, b, c) with a point Q = (x, y, z) are the entries of the vector 〈x− a, y− b, z − c〉. 1 Examples of vectors are the zero vector ~0 = 〈0, 0, 0〉, and the standard basis vectors ~i = 〈1, 0, 0〉,~j = 〈0, 1, 0〉, ~k = 〈0, 0, 1〉. Addition, subtraction and scalar multiplication of vectors can be done both geometrically and algebraically. The dot product ~v · ~ w between two vectors results is a scalar. It allows to define the length |~v| = √ ~v · ~v of a vector. The trigonometric cos-formula leads to the angle formula ~v · ~ w = |~v||~ w| cosα. By the CauchySchwarz inequality we can define the angle between two vectors using this formula. Vectors satisfying ~v · ~ w = 0 are called perpendicular. Pythagoras formula |~v + ~ w|2 = |~v|2 + |~ w|2 follows now from the definitions. Section 1.3: The cross product and triple scalar product After a short review of the dot product we introduce the cross product ~v × ~ w of two vectors ~v = 〈a, b, c〉 and ~ w = 〈p, q, r〉 in space. This new vector 〈br−cq, cp−ar, aq−bp〉 is perpendicular to both vectors 〈a, b, c〉 and 〈p, q, r〉. The product can be valuable for many things: it is useful for example to compute areas of parallelograms, the distance between a point and a line, or to 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 parallelepiped spanned by ~v and ~ w. Finally, we look at the triple scalar product (~u × ~v) · ~ w which is a scalar and is the signed volume of the parallelepiped spanned by ~u,~v and ~ w. Its sign tells about the orientation of the coordinate system defined by the three vectors. The triple scalar product is zero if and only if the three vectors are in a common plane. Section 1.4: Lines, planes and distances Because the 〈a, b, c〉 = ~n = ~u × ~v is perpendicular to ~x − ~ w if ~x, ~ w are both in the plane spanned by ~u and ~v, we are led to the equation ax + by + cz = d of the plane. Planes can be visualized by their traces, the intersection with coordinate planes as well as their intercepts, the intersection with coordinate axes. We often know the normal vector ~n = 〈a, b, c〉 to a plane and can determine the constant d by plugging in a known point (x, y, z) on equation ax + by + cz = d. We introduce lines by the parameterization ~r(t) = ~ OP + t~v, where P is a point on the line and ~v = 〈a, b, c〉 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 can be 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 a plane, the distance from a point to a line or the distance between two lines. We will also see how to compute distances between points, lines, planes, cylinders and spheres. Chapter 2. Curves and Surfaces Section 2.1: Functions, level surfaces, quadrics We first focus on functions f(x, y) of two variables. The graph of a function f(x, y) of two variables is defined as the set of points (x, y, z) for which z − f(x, y) = 0. We look at a few