Extended hour to hour syllabusOliver KnillMaths 21a, Summer 20041. Week: Geometry and Space29. June: Space, coordinates, distanceCoordinates for describing space was promoted by Descartes in the 16’th century at about thetime, when Harvard College was founded. A fundamental notion is the distance between twopoints. In order to get a feel about space, we will look at some geometric objects defined throughcoordinates. We will focus on circles and spheres and learn how to find the midpoint and radiusof a sphere given as a quadratic expression in x, y, z. This method is called completion of thesquare.30. June: Vectors, dot product, projectionsTwo points define an object which we call a vector. Vectors can be attached everywhere in spacebut are identified if they have the same length and direction. Vectors can describe for examplevelocities, forces or color. We learn first how to compute with vectors, use addition, subtractionand scaling both graphically as well as algebraically. The dot product is a product betweentwo vectors which results in a scalar. Using the dot product, we can compute length, angles orprojections.1. July: Cross product, linesThe cross product is a product between two vectors which results in a new vector perpendicularto both. The product can be used for many things. It is useful for example to compute areas,it can be used to compute the distance between a point and a line. It will also be important forconstructions like to get a plane through three points or to find the line which is in the intersectionof two planes. In general, there are different ways to describe a geometric object. For lines, wewill see the parametric description, as well as an implicit description which we will identify lateras the intersection between two planes.2. Week: Functions and Surfaces6. July: Planes, distance formulas1The simplest equations are linear equations. They describes planes. We will learn how todescribe planes using linear equations and how to construct them for example, from a line and apoint or from three points. As an application of the tools, we will look at some distance formulaslike the distance from a point to a plane, or the distance between two lines.7. July: Functions, graphs, quadricsFunctions of several variables play an essential role in this course. The graph of functions of twovariables define graphs z − f (x, y) = 0. We will also look at surfaces of the form g(x, y, z) = 0,where g is a function which only involves quadratic terms. These are called quadrics. Importantquadrics are spheres, ellipsoids, cones, cylinders as well as various hyperboloids.8. July: Implicit and parametric surfacesSurfaces can be described in two fundamental way. Implicitly or parametrically. The first formis g(x, y, z) = 0 like x2+ y2+ z2− 1 = 0 the second form 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 to go from oneform to the other like for the sphere, the plane, graphs of functions of two variables or surfaces ofrevolution. Using a computer, one can visualize surfaces very well. Computer algebra systemswith graphical capabilities are for the mathematician what the telescope is for the astronomer orthe microscope for the biologist. With a bit of patience you find your own surface which nobodyhas seen before.3. Week: Curves and Partial Derivatives13. July: Curves, velocity, acceleration, chain ruleCurves are one dimensional objects. Both in the plane as well as in space, they can take manydifferent forms. A special case are closed curves in space which are called knots. By differentiation,one obtains velocity and acceleration which are both vectors. The chain rule tells us how afunction changes along a curve.14. July: Arc-length, curvature, partial derivativesThere is a formula for the length of a curve. Lengths can be computed by evaluating a one-dimensional integral. The curvature of a curve is a quantity telling how much a curve is bent.Finally, we will see partial derivatives as well as see some partial differential equations abbre-viated as PDE’s.15. July: First midterm (on week 1-2)4. Week: Extrema and Lagrange Multipliers20. July: Gradient, linearization, tangentsThe gradient of a function is an important tool to describe the geometry of surfaces. Fundamentalis the property that the gradient vector ∇g is perpendicular to the implicit surface g = c. Thisallows us to compute tangent planes and tangent lines as well as to approximate a linearfunction by a linear function near a point. Many physical laws are actually just linearization ofmore complicated nonlinear laws.21. July: Extrema, second derivative testA central application of multi-variable calculus is to extremize functions of two variables. Onefirst identifies critical points, points where the gradient vanishes. The nature of these criticalpoints can be established using the second derivative test. There will be three fundamentallydifferent cases: local maxima, local minima as well as saddle points.22. July: Extrema with constraintsThe topic with maybe the most applications both in science or economics is to extremize a functionf(x, y) in the presence of a constraint g(x, y) = 0. A necessary condition for a critical point isthat the gradients of f and g are parallel. This leads to equations called the Lagrange equation.5. Week: Double Integrals and Surface Integrals27. July: Double integrals, type I,II regionsIntegration in two dimensions is first done on rectangles, then on regions bound by graphs offunctions. Similar than in one dimension, there is a Riemann sum approximation of theintegral. This allows us to prove results like Fubinis theorem on the change of the integrationorder. An application of double integration is the computation of area.28. July: Polar coordinates, surface areaMany regions can be described better in polar coordinates. Examples are so called roses whichtrace flower-like shapes in the plane but are graphs in polar coordinates. Changing coordinatescomes with an integration factor which can be explained also after introducing the surface area.29. July: Second midterm (on week 3-4)Triple Integrals and Line Integrals3. August: Triple integrals, cylindrical coordinatesTriple integrals allow the computation of volumes, moment of inertias or centers of masses ofsolids. First introduced for cubes it is then extended to more general regions bound by graphs offunctions of two variables. Some regions can
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