Extended hour to hour syllabusOliver KnillMaths 21a, Summer 20061. Week: Geometry and Space27. June: Space, coordinates, distanceClass starts with a short slide show highlighting some points of the syllabus and the material which waits foryou. Then we dive right into the subject. The idea to use coordinates to describe space was promoted byRen´e Descartes in the 16’th century at about the time, when Harvard College was founded. A fundamentalnotion is the distance between two points. Pythagoras theorem allows to measure a concrete distance in someBostonian unit. In or der to get a feel about space, we will look at some geometric objects defined by coordinates.We will focus on circles and spheres and lear n how to find the midpoint and radius of a sphere given as aquadratic expression in x, y, z. This method is c alled completion of the square. We will discuss, whatdistinguishes Euclidian distance from other distances. An other more philosophical q uestion is why our physicalspace is three-dimensional. A further topic fo r discuss ion is the existence of o ther coordinate systems like thephotographers coordinate system. Finally, we might mention GPS as an application of distance measurementor the open problem to find a perfect cube, a cube which the length of all sides, side dia gonals as well as spac ediagonals are integers and which will be a homework problem ...28. June: Vectors, dot product, projectionsTwo points P, Q define a vector~P Q. This includes the case P = Q, where~P Q is the null vector. Thevector connects the initial point P with the end point Q. Vectors can be attached everywhere in space but areidentified if they have the same length a nd dir e c tion. Vectors can describe for example velocities, forces orcolor or data. We learn first algebraic operations of vectors like addition, subtraction and scaling. This isdone both graphically as well a s algebraically. We introduce then the dot product between two vectors whichresults in a scala r. Using the dot product, we can compute length, angles and projections. By assumingthe trigonometric cos-formula, we prove the important formula ~v · ~w = |~v|| ~w| c os α, which relates length andangle with the do t product. This formula has some consequences like the Cauchy-Schwartz inequality orthe Pythagoras theorem. We mention the notation~i,~j,~k for the unit vectors.29. June: Cross product, linesThe third and las t lecture of the first week deals with the cross product of two vectors in space. The resultof this product in a new vector perpendicular to both. The product can be used for many things. It is usefulfor example to compute areas, it can be us e d to compute the distance between a point and a line. It willalso be important for constructions like to get a plane through three points or to find the line which is inthe intersection of two planes. The cross product is introduced as a determinant. We will prove the importantformula |~v × ~w| = |~v|| ~w| s in(α) and interpret it geometrically as an area of the parallelepiped spanned by ~v and~w. In general, there are different ways to descr ibe a geometric object. For lines, we will see a parametricdescription, as well as an implicit description. The later symmetric equation will later be identified as theintersection of two planes. The simplest equations are linear equations. A linear e quation ax + by + cz = c inthree variables geometrically defines a plane. This equation can be written as a(x−x0)+b(y −y0)+c(z −z0) = 0where (x0, y0, z0) is a point on the plane. The equation can be interpreted as the place which is perpendicular3to the vector ~n = ha, b, ci. We will then learn how to visualize a plane using traces and intercepts. A basicconstruction is to find the equation of a plane which passes through three points P ,Q, and R. As an application,we look at some distance formulas like the distance from a point to a plane, the distance from a pointto a line a s well as the the distance between two lines. These distance formulas are geometrically useful.They illustrate how one can use the dot and cross pro duct to measure in space.2. Week: Functions and Surfaces4. July: holiday5. July: Functions, graphs, quadricsAs the name ”multi-variable calculus” suggests, functions of several variables play an essential role in this course.In multivariable calculus, the focus is on functions of two or three variables. The graph of a function f(x, y)of two variables is defined as the se t of points (x, y, z) for which z − f (x, y) = 0. It is an example of a surface.After reviewing some conic sections in the plane like hyperbola and parabola, we will also look at implicitsurfaces of the form g(x, y, z) = 0, where g is a function which only involves quadratic terms. Surfaces of thistype are called quadrics. Important quadrics are spheres, ellipsoids, cones, cylinders, paraboloids andhyperboloids. You will have to know the names of these animals in the zoo of functions.6. July: Implicit and parametric surfacesSurfaces can be described in two fundamental ways: implicitly or parametrically. The implicit descrip-tion is g(x, y, z) = 0 like the sphere x2+ y2+ z2− 1 = 0, the parametric description is r(u, v) =(x(u, v), y(u, v), z(u, v)) like r(u , v) = (r cos(u) sin(v), r sin(u) sin(v), r cos(v)) In many cases, it is possible togo from one form to the other. There are four important types of surfaces for which o ne can do that: spheresor ellipsoids, planes, graphs of functions of two variables and surfaces of revolution. Using a computer, one canvisualize surfaces very well. Computer algebra systems with graphical capabilities can help to do so. Thesetools are fo r the mathematician what the telescope is for the astronomer or what the microscope is for thebiologist. With a bit of pa tience, you find your own surface which nobody has se e n before and which will bearyour name at the end of the course.3. Week: Curves and Partial Derivatives11. July: Curves, velocity, acceleration, chain ruleCurves are one-dimensional objects. One can look at curves both in the plane as well as in space, they can takemany different shapes. A special case are closed curves in space which are called knots. We will learn how todescribe curves by parametrization ~r(t) = hx(t), y(t), z(t)i. By differentiation, one obtains the velocity ~r′(t)and the acceleration ~r′′(t), which are both vectors. The speed of a curve at some p oint is the length |~r′(t)|of the velocity vector. The usual one
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