- Differentiability and the Tangent Plane References are to Salas & Hille, Calculus, 7th Edition
- 110.201 LINEAR ALGEBRA Fall 1997 Final Examination
- 110.202 CALCULUS III 23 February 2000 First Examination
- The Homotopy Extension Property This note augments material in Hatcher, Chapter 0.
- The Alexander-Whitney chain map The formula There is a well-known natural transformation ff, which consists of
- The Eilenberg-Zilber Theorem The chain complexes C(X xY ) and C(X) C(Y ) Throughout, we write C(X)
- Graded Algebra We reinterpret the concepts of cycles, cocycles, products etc., in terms of
- Coordinate Vectors References are to AntonRorres, 7th Edition
- Relations between Points and Sets Assume given a fixed metric space or topological space X and any subset E X.
- Attaching a 2-cell Compare Proposition 1.26 in Hatcher.
- 110.202 CALCULUS III 7 October 1998 First Examination
- 110.413 INTRODUCTION TO TOPOLOGY 27 February 2002 Mid-Term Examination
- Local Maxima and Minima References are to Salas/Hille/Etgen's Calculus, 8th Edition
- Inverse Function Theorem The contraction mapping theorem is a convenient way to prove existence theorems
- The Natural Exponential Function The factor log a in the derivative (see "General Exponential Functions")
- The Torus Triangulated Triangulation The most efficient way to triangulate the (2-dimensional) torus T
- 110.202 CALCULUS III 4 April 2000 Second Examination
- Canad. J. Math. Vol. 59 (6), 2007 pp. 11541206 k(n)-Torsion-Free H-Spaces and
- General Exponential Functions For fixed a, we want the exponential function ax
- The Riemann Zeta Function The Riemann zeta function is defined by the p-series
- Projections and Components Text: Salas/Hille/Etgen, Calculus Selected Chapters (Wiley, 1999), Eighth Edi-
- The Tangent Vector to a Curve Let C be a space curve parametrized by the differentiable vector-valued function
- Derivatives and Differentials References are to Salas/Hille/Etgen's Calculus, 8th Edition (Wiley, 1999)
- The Riemann Integral in Two Dimensions See also Step Functions in Two Dimensions, in this series.
- A Fubini Counterexample We attempt to evaluate the double integral
- Spanning and Linear Independence References are to AntonRorres, 7th edition
- Diagonalization References are to AntonRorres, 7th Edition
- Linear System Example: Repeated Real Root We solve Example 2 of 56 in [Simmons, Second edition], on p. 4312,
- Linear System Example: Complex Roots We solve Problem 1(h) of 56 in [Simmons, Second edition], on p. 433,
- Relations between Points and Subsets Assume given a metric or topological space X and any subset A X. We discuss
- Function Spaces In standard terminology,
- van Kampen's Theorem We present a variant of Hatcher's proof of van Kampen's Theorem, for the simpler
- Universal Coefficient Theorem for Cohomology We present a direct proof of the universal coefficient theorem for cohomol-
- 110.201 LINEAR ALGEBRA 8 October 1997 First Examination
- 110.201 LINEAR ALGEBRA 5 November 1997 Second Examination
- 110.202 CALCULUS III 23 February 2000 First Examination
- 110.202 CALCULUS III 4 April 2000 Second Examination
- 110.202 CALCULUS III April 2000 Second Examination
- 110.202 CALCULUS III 7 October 1998 First Examination
- 110.202 CALCULUS III 11 November 1998 Second Examination
- 110.202 CALCULUS III 11 December 1998 Final Examination
- 110.202 CALCULUS III 7 May 1998 Final Examination
- 110.202 CALCULUS III Spring 1998 Final Examination (Alternate B)
- 110.413 INTRODUCTION TO TOPOLOGY 2 March 2005 Mid-Term Examination
- 110.201 LINEAR ALGEBRA 17 December 1997 Final Examination
- 110.202 CALCULUS III 23 February 2000 Solutions to First Examination
- 110.202 CALCULUS III May 2000 Final Examination
- Row Space and Column Space References are to AntonRorres
- Graded Algebra We reinterpret the concepts of cycles, cocycles, products etc., in terms of
- The AlexanderWhitney chain map The formula There is a well-known natural transformation , which consists of
- The Homotopy Extension Property This note augments material in Hatcher, Chapter 0.
- Step Functions in Two Dimensions References are to Salas & Hille, Calculus, 7th Edition
- k(n)-torsion-free H-spaces and P(n)-cohomology J. Michael Boardman W. Stephen Wilson
- Pushouts and Adjunction Spaces This note augments material in Hatcher, Chapter 0.
- 110.413 INTRODUCTION TO TOPOLOGY 12 May 2004 Final Examination
- 110.202 CALCULUS III May 1998 Final Examination Alternate Edition
- 110.202 CALCULUS III Fall 1998 Final Examination
- Methods of Integration References are to Thomas & Finney, 8th edition.
- Inverse Linear Substitutions and Matrices We generalize Example 3 (on page 5 of AntonRorres) by solving the linear system
- First Order Linear Differential Equations The general first-order linear ordinary differential operator may be written for-
- Linear Substitutions and Matrix Multiplication This note interprets matrix multiplication and related concepts in terms of the
- Simplicial Complexes and -Complexes This note expands on some of the material on -complexes in 2.1 of Hatcher's
- Inverting 22 matrices In this note we invert the general 22 matrix as in Theorem 1.4.5 of Anton
- 110.202 CALCULUS III 23 February 2000 Solutions to First Examination
- Evaluation of Double Integrals The following Fubini-type theorem is fundamental to the evaluation of any Rie-
- 110.413 INTRODUCTION TO TOPOLOGY 14 May 2003 Final Examination
- The EilenbergZilber Theorem The chain complexes C(XY ) and C(X) C(Y ) Throughout, we write C(X)
- 110.202 CALCULUS III 11 November 1998 Second Examination
- Linear Transformations and Matrices References are to AntonRorres, 7th Edition
- 110.413 INTRODUCTION TO TOPOLOGY 3 March 2003 Mid-Term Examination
- 110.413 INTRODUCTION TO TOPOLOGY 2 March 2004 Mid-Term Examination
- 110.202 CALCULUS III 25 February 1998 First Examination
- The Natural Exponential Function The factor loga in the derivative (see "General Exponential Functions")
- Linear System Example: Repeated Real Root We solve Example 2 of x56 in [Simmons, Second edition], on p. 431-2,
- Universal Coefficient Theorem for Homology We present a direct proof of the universal coefficient theorem for homology
- The Riemann Integral in Two Dimensions See also Step Functions in Two Dimensions, in this series.
- 110.413 INTRODUCTION TO TOPOLOGY 12 May 2004 Final Examination
- Evaluation of Double Integrals The following Fubini-type theorem is fundamental to the evaluation of any Ri*
- 110.202 CALCULUS III Spring 1998 Final Examination (Alternate B)
- 110.202 CALCULUS III 25 February 1998 First Examination
- 110.202 CALCULUS III 7 October 1998 First Examination
- Linear System Example: Complex Roots We solve Problem 1(h) of x56 in [Simmons, Second edition], on p. 433,
- The Torus Triangulated Triangulation The most efficient way to triangulate the (2-dimensional) torus T
- 110.202 CALCULUS III 7 October 1998 First Examination
- 110.201 LINEAR ALGEBRA 5 November 1997 Second Examination
- Relations between Points and Subsets Assume given a metric or topological space X and any subset A X. We discuss
- Inverse Linear Substitutions and Matrices We generalize Example 3 (on page 5 of Anton-Rorres) by solving the linear sy*
- Inverting 2x2 matrices In this note we invert the general 2x2 matrix as in Theorem 1.4.5 of Anton-
- Universal Coefficient Theorem for Cohomology We present a direct proof of the universal coefficient theorem for cohomol-
- Pushouts and Adjunction Spaces This note augments material in Hatcher, Chapter 0.
- 110.202 CALCULUS III 4 April 2000 Second Examination
- 110.202 CALCULUS III 11 December 1998 Final Examination
- 110.201 LINEAR ALGEBRA 17 December 1997 Final Examination
- Attaching a 2-cell Compare Proposition 1.26 in Hatcher.
- Some Common Tor and Ext Groups We compute all the groups G H, Tor(G, H), Hom(G, H), and Ext(G, H),
- Universal Coefficient Theorem for Homology We present a direct proof of the universal coefficient theorem for homology
- The Real Projective Plane Triangulated Triangulation The most efficient way to triangulate the real projective plane RP2
- van Kampen's Theorem We present a variant of Hatcher's proof of van Kampen's Theorem, for the sim*
- 110.413 INTRODUCTION TO TOPOLOGY 14 May 2003 Final Examination
- 110.420 TOPOLOGY OF SURFACES March 1996 Mid-Term Examination
- 110.406 ANALYSIS II 3 March 1994 Mid-Term Examination
- 110.202 CALCULUS III 23 February 2000 Solutions to First Examination
- 110.420 TOPOLOGY OF SURFACES 14 May 1996 Final Examination
- Some Common Tor and Ext Groups Abstract
- 110.413 INTRODUCTION TO TOPOLOGY 2 March 2005 Mid-Term Examination
- Derivatives and Differentials References are to Salas/Hille/Etgen's Calculus, 8th Edition (Wiley, 1999)
- Linear System Example: Distinct Real Roots We solve Example 1 of x56 in [Simmons, Second edition], on p. 429,
- Differentiability and the Tangent Plane References are to Salas & Hille, Calculus, 7th Edition
- Step Functions in Two Dimensions References are to Salas & Hille, Calculus, 7th Edition
- First Order Linear Differential Equations The general first-order linear ordinary differential operator may be written*
- The Tangent Vector to a Curve Let C be a space curve parametrized by the differentiable vector-valued func*
- Spheres and Hopf Rings J. Michael Boardman
- Local Maxima and Minima References are to Salas/Hille/Etgen's Calculus, 8th Edition
- The Riemann Zeta Function The Riemann zeta function is defined by the p-series
- Derivatives Given a function f, assume that f0(a) = m, where a is fixed. We put
- Conditionally Convergent Spectral Sequences by J. Michael Boardman
- Linear Substitutions and Matrix Multiplication This note interprets matrix multiplication and related concepts in terms of *
- Function Spaces In standard terminology,
- The Real Projective Plane Triangulated Triangulation The most efficient way to triangulate the real projective plane *
- 110.202 CALCULUS III April 2000 Second Examination
- Diagonalization References are to Anton-Rorres, 7th Edition
- 110.202 CALCULUS III 4 April 2000 Second Examination
- 110.201 LINEAR ALGEBRA Fall 1997 Final Examination
- Row Space and Column Space References are to Anton-Rorres
- Linear System Example: Distinct Real Roots We solve Example 1 of 56 in [Simmons, Second edition], on p. 429,
- Derivatives Given a function f, assume that f (a) = m, where a is fixed. We put
- 110.202 CALCULUS III 4 May 2000 Final Examination
- Methods of Integration References are to Thomas & Finney, 8th edition.
- 110.413 INTRODUCTION TO TOPOLOGY 18 May 2005 Final Examination
- Simplicial Complexes and -Complexes This note expands some of the material on -complexes in x2.1 of
- Linear Transformations and Matrices References are to Anton-Rorres, 7th Edition
- 110.202 CALCULUS III 23 February 2000 First Examination
- 110.413 INTRODUCTION TO TOPOLOGY 15 May 2002 Final Examination
- 110.202 CALCULUS III 23 February 2000 Solutions to First Examination
- x0 Spectral sequences Conditionally Convergent Spectral Sequencesfirst or third quadrant of the p*
- 110.202 CALCULUS III 7 May 1998 Final Examination
- General Exponential Functions For fixed a, we want the exponential function ax to have at least the proper*
- Spanning and Linear Independence References are to Anton-Rorres, 7th edition
- 110.202 CALCULUS III 4 May 2000 Final Examination
- 110.413 INTRODUCTION TO TOPOLOGY 2 March 2004 Mid-Term Examination
- Projections and Components Text: Salas/Hille/Etgen, Calculus -Selected Chapters (Wiley, 1999), Eighth *
- 110.201 LINEAR ALGEBRA 8 October 1997 First Examination
- k(n)-torsion-free H-spaces and P (n)-cohomology J. Michael Boardman W. Stephen Wilson
- 110.202 CALCULUS III Fall 1998 Final Examination
- A Fubini Counterexample We attempt to evaluate the double integral
- Relations between Points and Sets Assume given a fixed metric space or topological space X and any subset E *
- 110.413 INTRODUCTION TO TOPOLOGY 27 February 2002 Mid-Term Examination
- 110.202 CALCULUS III May 1998 Final Examination Alternate Edition
- 110.202 CALCULUS III 11 November 1998 Second Examination
- 110.202 CALCULUS III 11 November 1998 Second Examination
- Inverse Function Theorem The contraction mapping theorem is a convenient way to prove existence theor*
- Field Coefficients Notation Let R be a ring with unit element 1. It need not be commutative (yet).
- 110.202 CALCULUS III May 2000 Final Examination
- Coordinate Vectors References are to Anton-Rorres, 7th Edition
- 110.413 INTRODUCTION TO TOPOLOGY 15 May 2002 Final Examination
- 110.413 INTRODUCTION TO TOPOLOGY 18 May 2005 Final Examination