Mathematics 8 - Spring 2000

Tentative Term Syllabus

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

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