MATH 4443/5443 - Introduction to Analysis I, Section 001 - Spring 2018
TR 10:30-11:45 a.m., 212 PHSC

Office Hours: Monday 12:30-1:30, Wednesday 11:00 a.m.-12:00 p.m., or by appointment, in 1101 PHSC.

First day handout

Course catalog description: Prerequisite: 4433. Integration of functions of a single variable. Series of real numbers. Series of functions. Differentiation of functions of more than one variable. No student may earn credit for both 4443 and 5443. (Sp)

Texts: All the books are feely available in pdf format for OU students from the OU library; OU students can purchase a cheap paper copy through the OU library:

• [A] Stephen Abbott, Understanding Analysis, Springer, 2nd ed., 2015
• [AK] Asuman G. Aksoy, Mohamed A. Khamsi, A Problem Book in Real Analysis, Springer, 2010
• [DD] Kenneth R. Davidson, Allan P. Donsig, Real Analysis and Applications, Springer, 2010
• [SV] Satish Shirali, Harkrishan Lal Vasudeva, Multivariable Analysis, Springer, 2011

Homework [read the following useful advice on writing proofs and advice from students]

• Homework 1, due January 25 (Thursday)
• Homework 2, due February 1 (Thursday)
• Homework 3, due February 8 (Thursday)
• Homework 4, due February 15 (Thursday)
• Homework 5, due February 27 (Tuesday) [please note the change in the due date!]
• Homework 6, due March 8 (Thursday)
• Homework 7, due March 15 (Thursday)
• Homework 8, due March 29 (Thursday)
• Homework 9, due April 5 (Thursday)
• Homework 10, due April 17 (Tuesday) [please note the unusual due date!]
• Homework 11, not to be turned in.

Tentative course content:

• Infinite series: definition, properties.
• Sequences and series of functions: uniform convergence of a sequence of functions, uniform convergence and differentiation, series of functions, power series, Taylor series, (Weierstrass Approximation Theorem).
• The Riemann integral: definition of Riemann integral, integrating functions with discontinuities, properties of the integral, the Fundamental Theorem of Calculus.
• Limits and continuity: limits of functions, continuous functions, Intermediate Value Theorem, (uniform continuity).
• Differentiation of functions of n variables: topology of Rⁿ, norms and inner products, derivative of a function of n variables.

Content of the lectures:

• Lecture 1 (Tue, Jan 16): Review: a brief review of some of the material from the handout "Summary of results from Analysis I" by Prof. Chavez-Dominguez - cardinality, supremum and infimum, completeness, Archimedean Principle, density of the rationals and the irrationals; definition of limit of a sequence (slightly modified): a_n→L if ∀ε>0 ∃N s.t. |a_n−L|<ε ∀n>N; an example of computing the limit of a concrete sequence directly from the definition.
• Lecture 2 (Thu, Jan 18): Review (cont.): more on limits: read parts G-L of the handout of Prof. Chavez-Dominguez and think about the proofs of the statements.
  The Monotone Convergence Theorem and a first look at infinite series: definition of increasing, decreasing, and monotone sequences; Monotone Convergence Theorem ([A], Theorem 2.4.2); (infinite) series, partial sums, convergence of a series; examples: telescoping series, proof that the series ∑_k k⁻² converges; proof of the divergence of the harmonic series [[A], pages 56-58 of Sec. 2.4]
  Reading assignment: read the Cauchy Condensation Test with its complete proof [[A], page 59 of Sec. 2.4]
• Lecture 3 (Tue, Jan 23): Properties of infinite series: Algebraic Limit Theorem for series; Cauchy Criterion for series; Comparison Test; geometric series; Absolute Convergence Test; Alternating Series Test; absolutely and conditionally convergent series; rearrangements; an example of a rearrangement of the alternating harmonic series that changes its sum to half of its sum; Riemann's Theorem of existence of a rearrangement of a conditionally convergent series that makes it converge to any number given in advance [[A], page 71-75 of Sec. 2.7]
• Lecture 4 (Thu, Jan 25): Properties of infinite series (cont.): any rearrangement of an absolutelu convergent series converges to the same limit (Theorem 2.7.10); Cauchy product of two series; an example of a (conditionally convergent) series whose Cauchy product with itself diverges; Mertens Theorem: if two series are convergent and (at least) one of them converges absolutely, then their Cauchy product converges [[A], page 75, 76 of Sec. 2.7]
• Lecture 5 (Tue, Jan 30): Uniform convergence of a sequence of functions: pointwise convergence of a sequence of functions; examples of computing the pointwise limits of functions; examples of sequences of differentiable functions whose limit is not differentiable or not even continuous; an unsuccessful attempt to prove a theorem that the pointwise limit of a sequence of continuous functions is continuous; modification of the definition of convergence of a sequence of functions - uniform convergence [[A], pages 173-177 of Sec. 6.2]
• Lecture 6 (Thu, Feb 1): Uniform convergence of a sequence of functions (cont.): more examples of sequences of functions and their limits; Cauchy criterion for uniform convergence (Theorem 6.2.5), Continuous Limit Theorem (Thereom 6.2.6); uniform continuity of a function - a motivating example (Example 4.4.3) and a definition (Definition 4.4.4); more facts about uniform convergence (Theorems 4.4.5 and 4.4.7); Cantor set and Cantor function.
  Just for fun: Banach-Tarski paradox (see also this video); a Math joke: question: "What's an anagram of Banach-Tarski?", answer: "Banach-Tarski Banach-Tarski." :)
  [[A], pages 177-179 and Example 6.2.12 on page 182 of Sec. 6.2; [A], pages 130-133 of Sec. 4.4; [A], pages 85, 86 of Sec. 3.1]
• Lecture 7 (Tue, Feb 6): Uniform convergence and differentiation: statement and proof of the Differentiable Limit Theorem (Theorem 6.3.1); examples and counterexamples [[A], Sec. 6.3]
  Series of functions: definition of poitnwise and uniform convergence of series of functions; Term-by-termp Continuity Theorem; Term-by-termp Differentiability Theorem; Cauchy Criterion for Uniform Convergence of Series; Weierstrass M-Test [[A], Sec. 6.4]
• Lecture 8 (Thu, Feb 8): Power series: defintion of a power series; convergence of a power series at x₀ implies its convergence at all x satisfying |x|<|x₀| (Theorem 6.5.1), so that the interval of convergence is symmetric about 0 (except maybe at the endpoints); radius of convergence of a power series; if a power series converges absolutely at x₀, then it converges uniformly on [−|x₀|,|x₀|] (Theorem 6.5.2); corollary: if R is the radius of convergence of a power series, then the power series is continuous on the interval (−R,R); Abel's Lemma (Lemma 6.5.3) [[A], pages 191-193 of Sec. 6.4]
• Lecture 9 (Tue, Feb 13): Power series (cont.): Abel's Theorem (Theorem 6.5.4); for a power series, pointwise convergence on a set A implies uniform convergence on any compact subset of A (Theorem 6.5.5); if a power series converges on (−R,R), then the term-by-term differentiated series also converges on (−R,R) (Theorem 6.5.6); summary of results on power series (Theorem 6.5.7)
  Taylor series: examples of Taylor series - formula for the sum of a geometric series, series obtained by differentiating the formula for the geometric series, series expansion of 1/(1+x²) and of arctan(x).
  [[A], pages 193-196 of Sec. 6.4; pages 197-199 of Sec. 6.5]
• Lecture 10 (Thu, Feb 15): Taylor series (cont.): a digression: the reason for the series of the "nice" function ƒ(x)=1/(1+x²) to converge only on (−1,1) is that if we consider the function ƒ(z)=1/(1+z²) as a function of the complex variable z, then ƒ(z) is not defined for z=±i; formula for the coefficients in the Taylor series (Theorem 6.6.2); Lagrange Remainder Theorem (Theorem 6.6.3); an example of application - estimating the number of terms needed to find the value of cos(x) on (−π,π] with a given tolerance; Taylor series centered at a≠0; an example of a function whose Taylor series does not converge to the function; analytic (C^ω) versus smooth (C^∞) functions [[A], pages 199-203 of Sec. 6.6]
• Lecture 11 (Tue, Feb 20): The definition of the Riemann integral: motivation of the concept of a definite integral; a partition of an interval; lower and upper sums of a function with respect to a partition; refinement of a partition; refining a partition increases the lower sums and decreases the upper sums (Lemma 7.2.3) [[A], Sec. 7.1, and pages 218, 219 of Sec. 7.2]
  A digression: Weierstrass Approximation Theorem, Bernstein's constructive approach to polynomial approximation, Bernstein polynomials; interpolation of functions: piecewise-linear interpolation, cubic spline interpolation; approximation of functions: linear regression, least squares; a Numerical Analysis course offered by the Math Department - MATH 4073.
• Lecture 12 (Thu, Feb 22): Lecture cancelled due to weather.
• Lecture 13 (Tue, Feb 27): The definition of the Riemann integral (cont.): the lower sum of ƒ for any partition is no greater than the upper sum ƒ for any other partition (Lemma 7.2.4); lower and upper integrals; lower integral of ƒ does not exceed the upper integral of ƒ (Lemma 7.2.6, without proof); definition of Riemann integrability; Integrability Criterion (Theorem 7.2.8); integrability of continuous functions (Theorem 7.2.9) [[A], pages 219-222 of Sec. 7.2]
• Lecture 14 (Thu, Mar 1): Exam 1 on Sections 2.4, 2.7, 6.2-6.6 (and parts of Sec. 3.1 and 4.4) covered in Lectures 1-10 and Homework assignments 1-5.
• Lecture 15 (Tue, Mar 6): Integrating functions with discontinuities: motivating examples showing how to deal with a finite number of finite jump discontinuities; Dirichlet's function (Example 7.3.3).
  Properties of the integral: additivity of domain (Theorem 7.4.1); linearity of definite integration (Theorem 7.4.2(i,ii)) [[A], pages 224-226 of Sec. 7.3 (omit the proof of Theorem 7.3.2); [A], pages 228-230 of Sec. 7.4]
• Lecture 16 (Thu, Mar 8): Properties of the integral (cont.): bounds on the integral, monotonicity, integrability of |ƒ| (Theorem 7.4.2(iii-v); defining integral over [a,b] when a>b, defining integral over [c,c] [[A], pages 229-231 of Sec. 7.4]
• Lecture 17 (Tue, Mar 13): Properties of the integral (cont.): Integrable Limit Theorem (Theorem 7.4.4) [[A], pages 231-232 of Sec. 7.4]
  The Fundamental Theorem of Calculus: motivation: if A(x₀) is the area under the graph of a continuous function ƒ between the vertical lines x=a and x=x₀ (with a<x₀), then the derivative A'(x₀) is approximately equal to the area under the graph of ƒ and between the vertical lines x=x₀ and x=x₀+Δx₀, divided by Δx₀, i.e., A'(x₀)≈ƒ(x₀); statement and proof of the first part of FTC (Theorem 7.5.1(i)) [[A], pages 234-235 of Sec. 7.5]
• Lecture 18 (Thu, Mar 15): The Fundamental Theorem of Calculus (cont.): statement and proof of the second part of FTC (Theorem 7.5.1(ii)); derivation of the formula for integration by parts as a consequence of the Leibniz rule (Exercise 7.5.6); derivation of the formula for change of variables in definite integrals as a consequence of the Chain Rule (Exercise 7.5.10) [[A], pages 234-238 of Sec. 7.5]
  A digression: differentiation of an integral depending on a parameter; computing integral of e^−x2 over R and using this result and the formula for differentiating an integral depending on a parameter in order to compute integrals of xⁿe^−x2 over R.
• Lecture 19 (Tue, Mar 27): Topological and metric spaces: topology, topological space; example: R with the "standard" topology (consisting of all open sets); convergence of a sequence in a topological space;
  metric ρ:X×X→[0,∞), metric space (X,ρ); example: R with ρ(x,y)=|x−y|, the unit sphere S² in R³ with ρ(x,y) given by the shortest distance between the points x,y∈S²; discrete metric d on X; if ρ is a metric on X, then ρ₁ defined by ρ₁(x,y)=ρ(x,y)/(1+ρ(x,y)) is also a metric; open ball B_r(x) in a metric space; definition of an open set in a metric space; every metric ρ on a set X induces topology on X; question: "when do two metrics ρ₁ and ρ₂ define the same topology on X?" - define equivalent metrics and equivalence classes on the set of metrics on X; convergence of a sequence in a metric space (X,ρ); closed sets (two definitions), closure of a set; Cauchy sequence in a metric space; complete metric space - in which every Cauchy sequence converges (to an element of the space); example: Q with the metric ρ(x,y)=|x−y| is not complete; example: (X,d) where d is the discrete metric - B_1/2(x)={x}, a sequence (x_n) converges to x if eventually each term of the sequence is equal to x, every subset of X is both open and closed, (X,d) is a complete metric space [[DD], pages 175-177 of Sec. 9.1]
• Lecture 20 (Thu, Mar 29): Topological and metric spaces (cont.): metrics ρ₁, ρ₂, ρ_∞, ρ_p for p∈[1,∞) in Rⁿ (considered as a set of points, not as a vector space); metrics ρ₁, ρ₂, ρ_∞, ρ_p for p∈[1,∞) in R^∞ (the set of infinite sequences); metric ρ_∞(ƒ,g)=sup_x∈[a,b]|ƒ(x)−g(x)| on the space C([a,b]) of continuous functions on [a,b], completeness of the metric space (C([a,b]),ρ_∞); metric ρ₁(ƒ,g) equal to integral of |ƒ(x)−g(x)| over [a,b] on the space C([a,b]) of continuous functions on [a,b]; an example demonstrating that the metric space (C([a,b]),ρ₁) is not complete; continuity of a map from the metric space (X,ρ) to the metric space (Y,τ) [[DD], Sec. 9.1]
  Normed vector spaces: norm || || on a vector space V; normed vector space (V,|| ||); norms || ||_p for p∈[1,∞), || ||_∞ on Rⁿ; sup norm on the set C(K) of continuous functions on a compact set K; a sequence (x_n) in a normed vector space (V,|| ||) is said to converge to x∈V if ||x_n−x||→0 as n→∞; a sequence (x_n) in (V,|| ||) is said to be Cauchy if for every ε>0 there exists an N such that ||x_n−x_m||<ε for every n,m≥N; a normed vector space (V,|| ||) is said to be complete if every Cauchy sequence converges (to an element of V); a Banach space is a complete normed vector space [[DD], pages 113-115 of Sec. 7.1, pages 117-118 of Sec. 7.2]
• Lecture 21 (Tue, Apr 3): Normed vector spaces (cont.): open ball B_r(x) in a normed vector space; an open set A in a normed vector space - such that for each element x∈A there exists a number r>0 such that B_r(x)⊂A; closed sets in a normed vector space; briefly about compact sets: a compact set K in a normed vector space is defined as a set such that every open cover of A has a finite subcover (or, equivalently, such that each sequence in K has a convergent subsequence); in finite-dimensional normed vector spaces a set is compact if and only if it is closed and bounded warning: in infinitely-dimensional normed vector spaces it is not true that A is compact if and only if it is closed and bounded - think about the sequence of vectors (x_n)=(0,0,...,0,1,0,...) (1 is at the nth position) in R^∞ that is in the unit ball but does not converge (it is not Cauchy);
  Cauchy-Schwarz inequality in Rⁿ; equivalence of any norm || || on any n-dimensional vector space V and the norm || ||₂ on Rⁿ (Theorem 7.3.1); if the norms || || and || ||' on the same vector space V are equivalent, then a sequence (x_n) in V converges in the norm || || exactly when it converges in the norm || ||' - this can be stated by saying that equivalent norms on a vector space V generate the same topology [[DD], pages 117-118 of Sec. 7.2, pages 120-121 of Sec. 7.3; pages 49-50 of Sec. 4.1]
  Inner product vector spaces: definition of inner product ⟨ , ⟩ on a vector space V; inner product vector space (V,⟨ , ⟩); norm associated with an inner product; Cauchy-Schwarz inequality in an inner product vector space; each positive definite n×n matrix (Q_ij) defines an inner product in Rⁿ; a nested sequence of spaces:
  {inner product vector spaces}⊂{normed vector spaces}⊂{metric spaces}⊂{topological spaces}
  [[DD], pages 124-125 of Sec. 7.4]
  Reading assignment: read Sections 7.1, 7.2, 7.3 (skip Corollaries 7.3.2, 7.3.4, and 7.3.5), and pages 124-125 of Section 7.4 of [DD].
• Lecture 22 (Thu, Apr 5): Inner product vector spaces (cont.): Cauchy-Schwarz inequality in a general inner product space; weighted inner product in the space C([a,b]) of continuous functions on [a,b] defined by using a weight function w:[a,b]→[0,∞) [[DD], pages 125-126 of Sec. 7.4]
  Linear functions between Euclidean spaces: linear maps betweeen linear spaces; matrix (a_αi) of a linear map A:U→V between the linear space U with basis e_i (i=1,2,...,dim(U)) and the linear space V with basis f_α (α=1,2,...,dim(V)): a_αi is the αth component of Ae_i in the basis f_α of V; endowing the space L(U,V) with a structure of a linear space; composition of linear operators [[SV], pages 32-35 of Sec. 2-3]
  Continuity: definitions of continuity of a map between topological spaces, of a map between metric spaces, and of a map between normed linear spaces; definition of continuity through limits of sequences, equivalence of the two definitions (Proposition 2-6.10); to establish discontinuity of a function ƒ at x, one can find two sequences (x_n) and (y_n) that converge to x, but such that ƒ(x_n) is different from ƒ(y_n) [[SV], pages 52, 54, 56 of Sec. 2-6]
  Reading assignment: read Corollaries 7.4.5, 7.4.5, and 7.4.7 on page 126 of Section 7.4 of [DD].
• Lecture 23 (Tue, Apr 10): Norm of a linear map: the operator norm of a linear map A:U→V between the normed linear spaces (U,|| ||), (V,|| ||'); main inequality: ||Ax||≤||A||x||; geometric meaning of ||A||; example: if A:Rⁿ→Rⁿ where Rⁿ is endowed with the norm || ||_∞, then the operator norm is ||A||= max_1≤j≤n∑_1≤i≤n|a_ij| (exercise: this is easy to prove - try to prove it!) [[SV], pages 62, 63 of Sec. 2-6]
  Derivatives of functions between normed spaces: reminder about the concepts of derivative, linearization, and differential of a function ƒ:R→R, geometric meaning of ƒ'(x); motivation and definition of the derivative of a map f:Rⁿ→R^m; example: computing the derivative of the map f:R²→R³ given by f(x)=f(x₁,x₂)=(1+3x₁, x₁²x₂, x₁²/(1−x₂)) - click here [[SV], pages 78, 79 of Sec. 3-2]
  Reading assignment: read Remark 3-2.2 and parts (a)-(c) of Example 3-2.3 of Shirali's book [[SV], pages 79-81 of Sec. 3-2]
• Lecture 24 (Thu, Apr 12): Derivatives of functions from Rⁿ to R^m: little o notation: a function r:R→R is said to be o(h) if r(h)/h→0 as h→0; more generally, a function r:Rⁿ→R^m is said to be o(h) if ||r(h)||/||h||→0 as h→0; examples: sin(h)−h=o(h), exp(h)−h−h=o(h), h^3/2=o(h), any smooth function whose Taylor expansion about 0 starts with h² is o(h);
  equivalent restatements of the definition of derivative of a function f:Rⁿ→R^m: we say that f is differentiable at x and that its derivative at x is Df(x)∈L(Rⁿ,R^m) if
  • f(x+h)=f(x)+Df(x)⋅h+||h||u(h), where u:R→R satisfies ||u(h)||→0 as h→0, or, equivalently, if
  • f(x+h)=f(x)+Df(x)⋅h+o(h) where o(h) stands for any function that is "little o" of h, or, equivalently, if
  • ||f(x+h)−f(x)−Df(x)⋅h||/||h||→0 as ||h||→0;
  discussion of the concepts of derivative and differential from Calculus I and their geometric meaning in the light of the new definition of derivative;
  proof of the product fule for the product of two R-valued functions [[SV], page 80 of Sec. 3-2]
  The Chain Rule and a corollary: statement and proof of the Chain Rule: if f:Rⁿ→R^m and g:R^m→R^p, then D(g∘f)(x)∈L(Rⁿ,R^p) is D(g∘f)(x)=Dg(f(x))⋅Df(x)
  [[SV], pages 85, 86 of Sec. 3-3]
  Reading assignment: read Examples 3-3.2(a,b) of Shirali's book [[SV], pages 86-88 of Sec. 3-3]
• Lecture 25 (Tue, Apr 17): The Chain Rule and a corollary (cont.): examples of application of the Chain Rule; a corollary of the Chain Rule: if E is a convex open subset of Rⁿ and f:E→R^m with ||Df(x)||₂≤M for each x∈E, then, for any a∈E and b∈E, we have ||f(b)−f(a)||₂≤M||b−a||₂ and ||f(b)−f(a)||_&infin≤n^1/2M||b−a||_∞ (Corollary 3-3.4)
  warning: the Mean Value Theorem is not valid for functions with values in R^m for m≥2, example of this: Problem 3-3.P9(a) [[SV], pages 86-88, 90, 95 of Sec. 3-3]
  Directional derivative: definition and examples of directional derivative D_bf(x) of a function f:E⊆Rⁿ→R^m, where b∈Rⁿ; expressing the directional derivative in terms of the (Frechet) derivative for a differentiable function: D_bf(x)=Df(x)⋅b; an important observation: if f is differentiable at x, then D_bf(x) is linear in b [[SV], page 82 of Sec. 3-2]
  Reading assignment (optional): read the proof of Corollary 3-3.4 of Shirali's book [[SV], pages 90-91 of Sec. 3-3]
• Lecture 26 (Thu, Apr 19): Exam 2 on Sections 7.1-7.5 of [A], Sections 4.1, 7.1-7.4, 9.1 of [DD], and Sections 2-6, 3-2, 3-3 of [SV], covered in Lectures 11, 13, 15-24 and Homework assignments 6-10.
• Lecture 27 (Tue, Apr 24): Partial derivatives: definition of partial derivatives: if e_j is the standard basis in Rⁿ, then the jth partial derivative of a function f:E⊆Rⁿ→R^m is defined as D_jf(x)=D_ejf(x); examples of calculations of partial derivatives; if the e_j (j=1,...,n) and λ_α (α=1,...,m) are the standard bases in Rⁿ and R^m respectively, and if the components of f:E⊆Rⁿ→R^m in the basis λ_α are the functions ƒ_α:E⊆Rⁿ→R, then the matrix elements of the derivative Df(x) are [Df(x)]_αj=D_ejƒ_α(x); Jacobian matrix of a function [[SV], pages 96-99 of Sec. 3-4]
• Lecture 28 (Thu, Apr 26): A digression: linearization of a function f:E⊆Rⁿ→R^m at a point a∈E: a function L_a:Rⁿ→R^m given by L_a(x)=f(a)+Df(a)⋅(x−a); geometric meaning of the linearization for a function ƒ:E⊆R→R; using linearization for approximate calculation of values of differentiable functions
  Second partial derivatives: for a function ƒ:E⊆Rⁿ→R, the first partial derivatives are D_iƒ:E⊆Rⁿ→R, and the second partial derivatives are defined as the partial derivatives of the first partial derivatives: D_ijƒ:=D_i(D_jƒ):E⊆Rⁿ→R; under certain conditions, the partial derivatives D_ijƒ and D_jiƒ are equal - see the statements of Schwarz's Theorem (Theorem 3-5.3) and Young's Theorem (Theorem 3-5.4); the Hessian Hess ƒ(x) of a function ƒ:E⊆Rⁿ→R at a point x∈E is the matrix of its second derivatives: Hess ƒ(x)=[D_ijƒ(x)];
  for a function f:E⊆Rⁿ→R^m with component functions ƒ_α:E⊆Rⁿ→R, one can similarly define the second partial derivatives as D_ijƒ_α:=D_i(D_jƒ_α);
  let f:E⊆Rⁿ→R^m, then the derivative of f at x∈E is the linear operator Df(x)∈L(Rⁿ,R^m), where L(Rⁿ,R^m)≅R^mn is the space of linear operators from Rⁿ to R^m), and, therefore, we can think of Df as a function Df:E⊆Rⁿ→L(Rⁿ,R^m)≅R^mn; the second derivative of f at x∈E is the derivative of the first derivative, i.e., the derivative of Df:E⊆Rⁿ→L(Rⁿ,R^m)≅R^mn, therefore D²f:=D(Df):E⊆Rⁿ→L(Rⁿ,L(Rⁿ,R^m))≅R^mn2; higher derivatives are defined inductively: D^kf:=D(D^k−1f) [[SV], pages 96-99 of Sec. 3-4]
• Lecture 29 (Tue, May 1): Taylor's Theorem for vector-valued functions of many variables: statement and proof of Taylor's Theorem for a function f:E⊆Rⁿ→R^m, rewriting the Taylor series in several different ways.
• Lecture 30 (Thu, May 3): Existence and uniqueness of solutions of ordinary differential equations: statement and proof of the Theorem of existence and uniqueness of solutions of ODEs based on the Contraction Mapping Theorem; an example of an initial-value problem without solution: dy/dx=ƒ(y), y(0)=0, where ƒ(y)=−1 if y≥0 and ƒ(y)=1 if y<0; an example of an initial-value problem with infinitely many solutions: dV/dt=V^2/3, V(0)=0 - this problem describes the formation and growth of water droplets in oversaturated vapor and the non-uniqueness is related to the fact that the droplet can start forming at any moment of time;
  finally, some math magic: see, e.g., the book Mathematics, Magic and Mystery of the Oklahoman Martin Gardner (1914-2010) or find another of the over 100 books written by him; the educational foundation Gathering for Gardner (G4G) which organizes bi-annual conferences for anybody excited about Math.
• Final Exam: Thursday, May 10, 8:00--10:00 a.m. in PHSC 212

Good to know:

• The greek_alphabet.
• Some useful notations.