PEI Math 521K study guide
PEI Math 521K · Work through your exam, one topic at a time.
Math 521K is a University of Prince Edward Island course in real analysis, and this page prepares you for its final assessment. It is aimed at mathematics majors and others who need a rigorous, proof-based grounding in single-variable analysis with a first look at metric spaces and multivariable differentiability. The PEI Math 521K study guide has 9 sections and about 3,707 words.
01Real Number System and Elementary Logic
This domain tests whether you can build proofs on the ordered-field and completeness axioms of R rather than on intuition, and whether you can manipulate quantifiers correctly. Expect items on supremum and infimum, the Archimedean property, density of the rationals, and the logical form of the ε-statements used throughout the course.
429 words, about 2 min02Sequences, Series, and Limits
This domain tests ε-N convergence, monotone and Cauchy criteria, subsequences and limsup, and the convergence tests for series with their exact hypotheses. Most lost marks come from applying a test outside its range of validity rather than from arithmetic.
395 words, about 2 min03Continuity and Uniform Continuity
This domain tests the ε-δ and sequential definitions of continuity, the global theorems available on compact intervals, and the distinction between continuity at each point and uniform continuity on a set. Expect both proofs and counterexamples on intervals that are open or unbounded.
425 words, about 2 min04Differentiation and Mean Value Theorems
This domain tests the derivative as a limit, the mean value theorems and their use in comparing functions, Taylor expansion with remainder, and the precise hypotheses needed for L'Hôpital's rule. Proofs that hinge on choosing the right auxiliary function are common.
352 words, about 2 min05Riemann and Improper Integration
This domain tests the Darboux formulation of the Riemann integral, integrability criteria, the fundamental theorem, and the careful limiting process that defines improper integrals at infinite limits or vertical asymptotes. For a partition P with mesh widths Δx i, define upper sums U(f, P) = Σ M i Δx i and lower sums L(f, P) = Σ m i Δx i.
410 words, about 2 min06Sequences and Series of Functions; Uniform Convergence
This domain tests the difference between pointwise and uniform convergence and the theorems that license interchanging limits with continuity, integration, and differentiation. Power series and the Weierstrass M-test are the main computational tools.
449 words, about 2 min07Metric Spaces, Topology of R^n, and Compactness
This domain tests open and closed sets, closure and interior, continuity through preimages, and compactness with its characterizations in R^n and in general metric spaces. Distinguishing results that hold only in R^n from those valid in every metric space is a recurring theme.
436 words, about 2 min08Multivariable Calculus: Derivatives, Inverse and Implicit Function Theorems
This domain tests the total derivative as a linear approximation, the relation between partial derivatives and differentiability, and the exact hypotheses of the inverse and implicit function theorems. Expect computations of Jacobians and implicit derivatives alongside hypothesis-checking questions.
421 words, about 2 min09Proof Technique and Rigorous Argumentation
This domain tests the standards a written proof must meet: correct quantifier order, justified transitions, valid induction, awareness of converse and contrapositive, and appropriate use of counterexamples. Graders reward explicit hypotheses checks and penalize gaps hidden by notation.
390 words, about 2 min