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HARVARD MATH 21A - Geometry and Space

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Extended hour to hour syllabusOliver KnillMaths 21a, Summer 20051. Week: Geometry and Space28. June: Space, coordinates, distanceClass starts with a short slide show highlighting some points of the s yllabus. Then we dive right intothe material. The idea to use coordinates to describe space was promoted by Ren´e Descartes inthe 16’th century at about the time, when Harvard College was founded. A fundamental notionis the distance between two points. We will use Pythagora s to measure a concrete distance insome Bostonian unit. In order to get a feel about space, we will look at some geometric objectsdefined through c oordinates. We will focus o n circles and spheres and learn how to find themidpoint and radius of a sphere given as a quadratic expression in x, y, z. This method is calledcompletion of the square. If time permits, we disc uss, what distinguishes Euclidian distancefrom other distances. An other more philosophical question is why our physical spac e is threedimensional. A further topic for discussion is the existence of other coordinate systems like thephotographvers coordinate system. Finally, we might mention GPS as an application of distancemeasurement. This will be a challenge problem.29 June: Vectors, dot product, projectionsTwo points P, Q define an object which we call a vector~P Q. The vector connects the initialpoint P with the end point Q. Vectors can be attached e verywhere in space but are identified ifthey have the same length and direction. Vectors can describe for example velocities, forces orcolor. We learn first algebraic operations of vectors like addition, subtraction and scaling. Thisis done both graphically as well as algebraically. We introduce then the dot product betweentwo vectors which results in a scalar. Using the dot product, we can c ompute length, anglesand projections. By assuming the trigonometic cos-formula, we prove the important formula~v · ~w = |~v|| ~w| cos α, which relates length and angle with the dot product. This formula has someconsequences like the Cauchy-Schwartz inequality or the Pythagoras theorem. We mentionthe notation~i,~j,~k for the unit vectors.30. June: Cross product, linesThe cross product of two vectors in space results in a new vector perpendicular to both. Theproduct can be used for many things. It is useful for example to compute areas, it can be usedto compute the distance between a point and a line. It will a lso be important for constructionslike to get a plane through three po ints or to find the line which is in the intersection of twoplanes. The cro ss product is introduced as a determinant. We will prove the important formula|~v × ~w| = |~v|| ~w| sin(α) and interpret it geometrically as an area of the parallelepiped spanned by ~v1and ~w. In general, there are different ways to describ e a geo metr ic object. For lines, we will seethe parametric description, as well as an implicit description which we will be identified later asthe intersection of two planes.2. Week: Functions and Surfaces5. July: Planes, distance formulasThe simplest equations are linear equations. A linear e quation ax + by + cz = c in s pace definesa plane. This equation can be written as a(x − x0)+ b(y − y0)+ c(z −z0) = 0 where (x0, y0, z0) is apoint on the plane and interpreted as the placne which is perp e ndicula r to the vector ~n = (a, b, c).We will then learn how to visualize a plane using traces and intercepts. A basic constructio n isto 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 froma point to a line or the distance between two lines.6. July: Functions, graphs, quadricsAs the name ”multivariable calculus” suggests, functions of several variables play an essential rolein this course. The graph of functions o f two variables is defined as the set of points (x, y, z)for which z − f (x, y) = 0. After reviewing some conic sections, we will also look at surfacesof the form g(x, y, z) = 0, where g is a function which only involves quadratic terms. These arecalled quadrics. Important quadrics are spheres, ell ipsoids, cones, cylinders as well as va rioushyperboloids.7. July: I mplicit and parametric surfacesSurfaces can be des c ribed in two fundamental ways: 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 surface s very well. Computer algebra systemswith graphical capabilities are for the mathematician what the telescope is for the astronomer orthe microscope for the biolog ist. With a bit of patience you find your own surface which nobodyhas seen before.3. Week: Curves and Partial Derivatives12. 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 sp e c ial 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.13. 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.14. July: First midterm (on week 1-2)4. Week: Extrema and Lagrange M ultipliers19. July: Gradient, linearization, t angentsThe gradient of a function is an important tool to describe the geometry of surfaces. Funda mentalis 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.20. July: Extrema, second derivative testA central application of multi-variable calculus is to extremize functions of two variables. Onefirst identifies critical points, p oints where the gradient vanishes. The


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HARVARD MATH 21A - Geometry and Space

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