These notes are intended to supplement the texts. They mostly provide alternate treatments of selected topics and cover some topics that are barely addressed, if at all, in the texts. Some of them give worked examples. They are not required reading. Some people will find them more helpful than others.

Most notes are available in both DVI and PDF formats. The DVI version is more reliable (being the original), but is readable only on systems (usually large) that support TeX. The PDF version is widely readable, but certain fonts may be incorrect or missing. Either version may be affected by the quality of the fonts used by your local system. Try both to see which works better for you. Hard copies are also available from me (JMB) on request.

Notes for 110.108 Calculus I

• Derivatives (2 pages) - available in your choice of DVI format or PDF format.
  Differentiability of a function of one variable is defined as the existence of a good linear approximation. Proofs in this context of the standard rules for derivatives are given.
• General Exponential Functions (2 pages) - available in your choice of DVI format or PDF fornat.
  The general exponential function, a to the power x, is introduced axiomatically, and its algebraic properties are deduced. When we attempt to differentiate it, we find the natural logarithm of a and deduce its properties.
• The Natural Exponential Function (1 page) - available in your choice of DVI format or PDF format.
  This is presented as a special case of the general exponential function. Its properties follow easily.

Notes for 110.109 Calculus II

• Methods of Integration (3 pages) - available in your choice of DVI format or PDF format.
  This is a summary of the main methods of integration.
• The Riemann Zeta Function (2 pages) - available in your choice of DVI format or PDF format.
  The p-series for the Riemann zeta function is discussed as an example of a series that converges too slowly for practical computation.

Notes for 110.201 Linear Algebra

• Linear Substitutions and Matrix Multiplication (2 pages) - available in your choice of DVI format or PDF format.
  Matrix multiplication is explained by an example in terms of composition of linear substitutions. This context leads naturally to the associative law and the identity and zero matrices.
• Inverse Linear Substitutions and Matrices (1 page) - available in your choice of DVI format or PDF format.
  The inverse of a matrix is explained by an example in terms of solving a linear system, or equivalently as row reduction applied to the augmented matrix [A|I].
• Inverting 2x2 Matrices (2 pages) - available in your choice of DVI format or PDF format.
  The generic 2x2 matrix is inverted using only standard row reduction techniques, without recourse to guesswork or ingenuity.
• Coordinate Vectors (2 pages) - available in your choice of DVI format or PDF format.
  A basis B of a general vector space V is described as a method of identifying V with euclidean n-space. This leads to the formula for the effect of a change of basis on coordinate vectors. When V is an inner product space, one sees why B should be chosen orthonormal.
• Spanning and Linear Independence (4 pages) - available in your choice of DVI format or PDF format.
  The concepts of spanning set, linear independence and basis are compared.
• Row Space and Column Space (2 pages) - available in your choice of DVI format or PDF format.
  A worked example computes the null space, row space, rank and column space of a 4x5 matrix A.
• Linear Transformations and Matrices (2 pages) - available in your choice of DVI format or PDF format.
  Given bases of vector spaces V and W, the matrix of a general linear transformation from V to W is obtained. Composition of linear transformations corresponds to matrix multiplication. The formula for change of bases is obtained.
• Diagonalization (2 pages) - available in your choice of DVI format or PDF format.
  This discusses the effect of using a non-standard basis on the matrix of a linear transformation from euclidean n-space to itself. The matrix is diagonalizable if a basis of eigenvectors exists.

Notes for 110.202 Calculus III

• Projections and Components (1 page) - available in your choice of DVI format or PDF format.
  An arbitrary vector is decomposed as a sum of a vector that is parallel to a given nonzero vector and one that is orthogonal (perpendicular) to it. This leads to a short proof of Schwarz's inequality.
• The Tangent Vector to a Curve (1 page) - available in your choice of DVI format or PDF format.
  The derivative of a vector-valued function is identified with the geometric tangent vector to the curve traced out.
• Derivatives and Differentials (2 pages) - available in your choice of DVI format or PDF format.
  The directional derivative of a scalar function f(x) of a vector variable x is introduced first, and used to define the differential df. Partial derivatives are treated as a special case. The coordinate differentials are introduced last.
• Differentiability and the Tangent Plane (2 pages) - available in your choice of DVI format or PDF format.
  The differentiability of a function of two variables is discussed, both analytically and geometrically, in terms of the tangent plane.
• Local Maxima and Minima (2 pages) - available in your choice of DVI format or PDF format.
  The various possible behaviors of a scalar function f(r) of a 2-dimensional vector r near a stationary point are classified in terms of the quadratic form Q(h).
• Step Functions in Two Dimensions (2 pages) - available in your choice of DVI format or PDF format.
  The elementary properties of step functions in 2 dimensions are discussed, as a prelude to the Riemann integral.
• The Riemann Integral in Two Dimensions (2 pages) - available in your choice of DVI format or PDF format.
  The Riemann integral is defined in terms of step functions and the order-preserving property. This leads to a short proof of Fubini's Theorem, with no mention of continuity.
• Evaluation of Double Integrals (2 pages) - available in your choice of DVI format or PDF format.
  The main result is a Fubini-type theorem, based on the Riemann integral defined as the limit of Riemann sums, with no mention of continuity, A short sketch proof is given. Also listed are two auxiliary theorems that are needed to make the result useful for applications.
• A Fubini Counterexample (1 page) - available in your choice of DVI format or PDF format.
  This example shows that the order of integration in a double integral cannot be reversed if the integrand has a bad discontinuity, even if all the single integrals involve only continuous functions.

Notes for 110.302 Differential Equations

• First Order Linear Differential Equations (2 pages) - available in your choice of DVI format or PDF format.
  An example is solved using the method of variation of a parameter. This is claimed to be cleaner and easier than the standard approaches, and involves no guesswork.
• Basic Laplace Transforms (1 page) - available in your choice of DVI format or PDF format.
• Linear System: Distinct Real Roots (1 page) - available in your choice of DVI format or PDF format.
  A linear system with constant coefficients whose auxiliary equation has distinct real roots is solved using Laplace transforms.
• Linear System: Repeated Real Root (1 page) - available in your choice of DVI format or PDF format.
  A linear system with constant coefficients whose auxiliary equation has a repeated real root is solved using Laplace transforms.
• Linear System: Complex Roots (2 pages) - available in your choice of DVI format or PDF format.
  A linear system with constant coefficients whose auxiliary equation has complex roots is solved, first by the traditional method, and second by Laplace transforms. The second uses no complex numbers.

Notes for 110.405 Analysis I also 110.413 Introduction to Topology

• Relations between Points and Sets (1 page) - available in your choice of DVI format or PDF format.
• Relations between Points and Subsets (1 page) - available in your choice of DVI format or PDF format.
  These pages display all the possible relations between a point p and a subset E or A of a topological space. The second version is only slightly different. At the first level of information, no distinction is made between p and other points of its neighborhood; this defines the interior, exterior and boundary and hence open and closed sets. The second level of information allows the definition of limit points and isolated points.
• Function Spaces (2 pages) - available in your choice of DVI format or PDF format.
  This is a brief introduction to the compact-open topology.
• Inverse Function Theorem (2 pages) - available in your choice of DVI format or PDF format.
  A proof of the inverse function theorem of Calculus III by means of the contraction mapping theorem.

Notes for 110.615-616 Algebraic Topology

• Pushouts and Adjunction Spaces (4 pages) - choose
  • Pushouts in DVI format or
  • Pushouts in PDF format.
  
  This note defines pushouts and adjunction spaces in general, and reviews their elementary properties, with many examples. Most of the constructions in Chapter 0 of Hatcher's Algebraic Topology are examples of adjunction spaces.

• Attaching a 2-cell (2 pages) - available in your choice of DVI format or PDF format.
  This note applies van Kampen's Theorem to compute the effect on the fundamental group of attaching a 2-cell to a space. Compare Proposition 1.26 in Hatcher.

• van Kampen's Theorem (3 pages) - choose
  • van Kampen, in DVI format or
  • van Kampen, in PDF format.
  
  This note presents an alternate proof of van Kampen's Theorem from the pushout point of view, for the case where the space is covered by two open sets.

• The Torus Triangulated (2 pages) - choose
  • Torus triangulated, in DVI format or
  • Torus triangulated, in PDF format.
  
  The most efficient triangulation of the torus is presented. Explicit cycles and cocycles are found whose classes generate the homology and cohomology. These are used to compute cup products simplicially.

• The Real Projective Plane Triangulated (2 pages) - choose
  • RP2 triangulated, in DVI format or
  • RP2 triangulated, in PDF format.
  
  The most efficient triangulation of the real projective plane is presented. By removing one simplex, one obtains the Moebius band. The Klein bottle is displayed as the connected sum of two copies of the real projective plane.

• Simplicial Complexes and Delta-Complexes (4 pages) - available in your choice of DVI format or PDF format.
  This note compares simplicial complexes, ordered simplicial complexes, and delta-complexes.
• Some Common Tor and Ext Groups (6 pages) - choose
  • Tor and Ext groups, in DVI format or
  • Tor and Ext groups, in PDF format.
    This note computes all the groups G tensor H, Tor(G,H), Hom(G,H) and Ext(G,H) for G and H any of Z, Z/n or Q, including the difficult case Ext(Q,Z).
• Field Coefficients (2 pages) - available only in DVI format.
  The easy case of the Universal Coefficient Theorem when the ground ring is a field is discussed, starting from integer coefficients as in Munkres' book.
• Homotopy Groups of Spheres (1 page) - available in your choice of DVI format or PDF format.
  One pageful of homotopy groups of spheres. It only goes up to the 12-sphere and the 23rd homotopy group.
• Universal Coefficient Theorem for Homology (2 pages) - choose
  • universal coefficient theorem for homology, in DVI format or
  • universal coefficient theorem for homology, in PDF format.
    This note presents a direct proof of the universal coefficient theorem for homology that is simpler and shorter than the standard proof.
• Universal Coefficient Theorem for Cohomology (2 pages) - choose
  • universal coefficient theorem for cohomology, in DVI format or
  • universal coefficient theorem for cohomology, in PDF format.
    This note presents a direct proof of the universal coefficient theorem for cohomology which is essentially dual to the proof for homology.