The following is a tentative
syllabus for the course. This page will be updated irregularly. On
the other hand, the weekly syllabus contained in the Homework Assignments page will always be
accurate.
Lectures | Sections in Text | Brief Description |
---|---|---|
Day 1: 3/27 | Bits of 17.1, 17.5 | General Notions of ODEs, some examples, Slope Fields |
Day 2: 3/29 | 3.4, 17.2 | Separable Equations and Applications (simple mixing, growth, cooling) [No homogeneous equations] |
Day 3: 3/31 | 6.1 | Integration by Parts |
Day 4: 4/3 | 7.9 (17.4) | General First Order Linear DEs: Motion in a viscous medium, general mixing problems |
Day 5: 4/5 | 17.1, 3.7 | Derive DE for a oscillating spring with resistance [harmonic motion] as an example of a second order constant coefficient DE (with nontrivial linear term). How to find the solutions? Discuss differential operators and properties of linearity (Thms 1 and 2 of 17.1). Show how an nth order equation can be reduced to a system of first order equations. |
Day 6: 4/7 | 17.6, 17.7 (part) | Second order constant coeff DE (real roots). With notion of a system first order equations, and that homogeneous solutions form a vector space, discuss dimension of space, and notion of linear independence of solutions. Apply to second order real roots case. |
Day 7: 4/10 | Appendix I, 17.7 (part) | Complex numbers, complex exponential, extracting roots, second order constant coefficient DE with complex roots |
Day 8: 4/12 | 7.1 | Volumes of Revolution (disks: brief review; mostly on shells) |
Day 9: 4/14 | 4.8 | Taylor polynomials, pictures, start error term (error for a fixed degree approximation) |
Day 10: 4/17 | 9.5, 9.8 | Error estimates with Taylor polynomials [what degree is necessary to guarnatee an error less that epsilon], ratio test, radius of convergence and center |
Day 11: 4/19 | 9.2, 9.6 | Taylor series, manipulation of series (e.g. $\int(\sin x / x)\, dx$ or $\sin(3x)$ from $\sin x$, geometric series; prove Euler's formula via series |
4/19 | First Hour Exam | |
Day 12: 4/21 | 10.1, 10.2(start) | R^n coordinates and vector space properties. Distance formula, coordinate planes, simple graphs, algebraic and geometric interpretation of vector space operations. |
Day 13: 4/24 | 10.2 finish | Dot products and projections |
Day 14: 4/26 | 10.3 | Cross products and geometry: RH rule, characterization via 3x3 determinants, |u x v| is the area of a parallelogram, |u * (v x w)| is the volume of a parallelepiped. |
Day 15: 4/28 | 10.4 | Lines in R^n: parametric form, vector form, standard form, intersecting parallel, and skew lines. Derive the equation of a plane in standard and vector form. |
Day 16: 5/1 | 10.4 | Planes in R^n: More examples of how to determine a plane (3 noncolinear points, two lines, point and a line...), angle between planes, line of intersection of two planes, distance from a point to a plane. |
Day 17: 5/3 | 10.6 + ? | Systems of linear equations --- natural generalization of finding line of intersection of two planes, matrix representation of linear systems, solving systems via elementary operations, equivalent systems and rowspace, row reduction, echelon form |
Day 18: 5/5 | 10.6 + ? | AX = 0 with m < n yields infinitely many solutions (dimension of a solution space from rank of matrix). Matrix algebra. Note that x |--> Ax is a linear maps. Relationship between solutions to homogeneous and inhomogeneous systems. Comparison to systems of linear differential equations. |
Day 19: 5/8 | 10.6 + ? | Show Ax = b is solvable iff b is in column space of A; note that column space is a vector space of rank equal to rank of matrix. Invertibility and connection to solvability of Ax=b, Ax = 0. Perhaps inverses via row reduction (i.e. as a product of elementary matrices) |
Day 20: 5/10 | 12.1, 12.2 | Functions of several variables, limits, and continuity |
5/10 | Second Hour Exam | |
Day 21: 5/12 | 12.3, 12.4 | Partial derivatives |
Day 22: 5/15 | 12.3, 12.4 | Tangent Planes and normal lines |
Day 23: 5/17 | 12.5 | The chain rule (no higher order, homogeneous functions, or polar) |
Day 24: 5/19 | 12.7 | Gradients and Directional Derivatives |
Day 25: 5/22 | 12.7 | Gradients and Directional Derivatives |
Day 26: 5/24 | 13.1 | Extreme Values on open sets (no quadratic forms) |
Day 27: 5/26 | 13.2, 13.3 | Extreme Values on restricted domains, Lagrange Multipliers |
Day 28: 5/30 (x-hour) | 13.3 | Lagrange Multipliers, loose ends |
Last modified by
P. Kostelec on 24 Apr 2000