Faculty of Arts and Science

Polynomials and rational functions, trigonometry, exponential and logarithmic functions, inequalities, rudiments of probability and counting.

Note: |
This course may not be taken for credit by students with Mathematics 30‑1 or Pure Mathematics 30 |

Linear systems. Vectors and matrices. Determinants. Orthogonality and applications. Vector geometry. Eigenvalues, eigenvectors, and applications. Complex numbers.

Differentiation of elementary functions, the chain and product rules, extrema problems, integration. Applications from management, humanities and the social sciences.

Note: |
Mathematics 1510 may not be counted toward the requirements for a major in Mathematics and is not suitable for students requiring more than one semester of Calculus. |

Functions. Limits. Continuity. Differentiation and integration of polynomial, rational, root, trigonometric, exponential, and logarithmic functions. Applications of derivatives, including linear approximations and Taylor polynomials. Curve sketching and optimization. Anti-derivatives. Change of variable. Definite integrals. Fundamental Theorem of Calculus.

Recommended Background: |
Mathematics 31 and a blended grade of at least 75 percent in Mathematics 30‑1 or Pure Mathematics 30 |

Logic, proofs. Set theory. Relations and functions. Finite and countable sets. Induction. Examples of axiomatic mathematical theories.

Principles of Logic. Number Systems and Bases. Sets of real numbers: Integers, Rationals, Irrationals. Modular Arithmetic and applications. Divisibility, primes and elementary number theory.

Note: |
Students should not take Mathematics 2090 if they have received credit for Mathematics 2000 prior to enrolling in Mathematics 2090. |

Applications of integration; logarithmic, exponential, and hyperbolic functions; inverse functions; inverse trigonometric and hyperbolic functions; indeterminate forms; improper integrals; techniques of integration; polar coordinates; introduction to differential equations.

Sequences and series, convergence tests, Taylor's series, vector-valued functions of a real variable, applications to analytic geometry, partial derivatives.

Calculus of functions of several variables: partial differentiation, chain rule, applications, multiple integration, change of variables, theorems from vector analysis, including Stokes’ Theorem.

First Order Logic. Validity, provability, completeness, consistency, independence, categoricity, decidability, Gödel’s Theorem.

Introduction to classical geometry from the axiomatic point of view. Lines and affine planes. Separation, order, similarity, congruence. Isometries and their classification. Groups of symmetries. Projective, hyperbolic and inversive geometries.

Groups, abelian groups, subgroups, quotient groups. Homomorphism. Isomorphism theorems. Lagrange’s theorem. Permutation groups. Sylow theorems. Commutative rings, subrings, ideals. Quotient rings and ideals. Polynomial rings.

Vector spaces over the real and complex numbers. Basis and dimension. Linear transformations. Change of basis. Gram-Schmidt orthogonalization. Eigenvectors and diagonalization. Canonical forms. Cayley-Hamilton Theorem.

Division algorithm. Fundamental Theorem of Arithmetic. Euclidean Algorithm. Linear Diophantine equations. Congruences. Chinese Remainder Theorem. Quadratic reciprocity. Additional topics such as Pythagorean triples, Gaussian integers, sums of squares, continued fractions, arithmetic functions, or cryptography.

Rigorous treatment of the notions of calculus of a single variable, emphasizing epsilon-delta proofs. Completeness of the real numbers. Upper and lower limits. Continuity. Differentiability. Riemann integrability.

Complex number system and complex plane. Analytic functions. Complex integration. Power series. Calculus of residues.

First order ordinary differential equations. Second and higher order ordinary differential equations. Linear systems of ordinary differential equations. Qualitative theory of ordinary differential equations. Applications. Series solutions. Singular point expansions. Elementary linear difference equations.

Adjoints. Oscillation theory. Matrix methods. Matrix exponential functions. Sturm-Liouville theory. Orthonormal systems and Fourier series. Eigenfunction expansions. Laplace, Fourier and Mellin transforms. Convolutions. Convergence theory. Plancherel and Parseval formulae. Distributions. Solving PDEs using integral transforms. Fundamental solutions. Separation of variables. Heat, wave and Poisson equations. Harmonic functions.

Topological spaces. Topology of metric spaces. Continuity. Open covers and compactness. Separation. Connectedness.

Polynomial rings. Fields and field extensions, construction problems. Finite fields. Galois Theory. Fundamental Theorem of Algebra.

Sequences and series of functions. Uniform continuity. Uniform convergence. The Stone-Weierstrass Theorem. The Lebesgue (or Riemann-Stieltjes) integral. Fourier series. Other topics.