Syllabus for Multivariable Calculus (Math 211)
             Elementary vector analysis:
               Scalar or dot product
               Cross or vector product
               Lines and planes
               Differentiation of vector-valued functions
               Tangent vectors
               Velocity and acceleration
             Definitions involving functions of several variables:
               Limit
               Continuity
               Differentiability
               Partial derivative
               Directional derivative
               Gradient
               Tangent plane to a surface
             Computations using the usual rules of differentiation:
               Partial derivatives
               Directional derivatives
               Gradients
               Tangent planes to surfaces
             Computations and proofs using the definitions:
               Prove continuity (or lack thereof) of a function at a point
               Compute partial derivatives and directional derivatives from the definition
             Maxima and minima of functions of several variables:
               Finding critical points
               The second derivative test for local maxima/minima and saddle points
               The method of Lagrange multipliers
             Double integrals:
               Cartesian and polar coordinates
               Finding area, surface area and volume
             Triple integrals:
               Cartesian, cylindrical and spherical coordinates
               Finding volume and mass
             Line integrals:
               Basic properties
               Green’s Theorem
             Syllabus for Linear Algebra (Math 271 and 272)
             Basic definitions:
               Vector space
               Subspace
               Span of a subset
               Linear independence
               Basis and dimension
               Linear transformation
               Kernel or null space
               Image or range
               Inverse of a matrix or linear transformation
               Determinant and trace
               Characteristic polynomial
               Eigenvalues and eigenspaces
               Diagonalizability
               Similarity
             Computational techniques:
               Determine when a subset is a subspace
               Basic matrix manipulations
               Row operations on matrices
               Solving systems of linear equations
               Find the inverse of a matrix
               Find a basis of a given subspace
               Find the nullity, rank, trace, and determinant of a matrix
               Find the null space N(T) and range R(T) of a linear transformation T
               Given bases of V and W, find the matrix of a linear transformation T : V → W
             Given a matrix or linear transformation:
               Compute its characteristic polynomial
               Find its eigenvalues and eigenspaces
             Basic results to know:
               dim N(T) + dim R(T) = dim V
               nullity(A) + rank(A) = number of columns of A
               Criteria for A−1 to exist
               Criteria for A to be diagonalizable
             Write simple proofs of problems involving subspaces, linear maps, linear indepen-
             dence, spanning sets, null spaces and ranges.
             Syllabus for Algebra (Math 350)
             Sets and mappings:
               One-to-one, onto and bijective maps
               Equivalence relations and equivalence classes
             Groups:
               Uniqueness of identities and inverses
               The order of an element
             Subgroups:
               Cosets
               Lagrange’s Theorem
               Normal subgroups
               Quotient groups
             Group homorphisms:
               Kernels and images
               Isomorphisms
               The basic homomorphism theorem: G/Ker(ϕ) ≃ Im(ϕ)
             Permutations:
               Sn and cycle decomposition
               Transpositions and An
             Rings:
               Commutative rings
               Rings with unit element
               Fields
               Integral domains
             Ideals:
               Left and right ideals
               Two-sided ideals
               Quotient rings
             Ring homomorphisms:
               Kernels and images
               Isomorphisms
               The basic homomorphism theorem: R/Ker(ϕ) ≃ Im(ϕ)
             Quotient rings and fields:
               Criteria for R to be a field
               Maximal ideals
               Criteria for R/M to be a field
             Polynomial rings k[x], k a field:
               The division algorithm
               Every ideal in k[x] is principal
               Irreducible polynomials and unique factorization
               Maximal ideals in k[x]
             Definitions and statements of the basic theorems are important
             Syllabus for Analysis (Math 355)
             The real numbers:
               Mathematical induction
               Rational and irrational numbers
               Real numbers and the completeness axiom
             Sequences:
               The convergence of bounded, monotone sequences of reals
               Cauchy sequences
               Bolzano-Weierstrass Theorem (for bounded sequences)
             Point-set theory:
               Countable and uncountable sets
               Accumulation points (also called cluster points or limit points)
               Bolzano-Weierstrass Theorem (for bounded infinite sets)
               Open and closed sets
               Compact sets
               Heine-Borel Theorem
             Continuous functions:
               Intermediate Value Theorem
               Basic properties of functions continuous on a compact set:
                Boundedness
                Attainment of extreme values
                Uniform continuity
               Continuity of sums, products, quotients of continuous functions
             Differentiability and derivatives:
               Limit definition of derivative
               Derivatives at local extreme points
               Rolle’s Theorem, Mean Value Theorem
             Integration:
               Definition of Riemann integral
               Integrability of a continuous function over [a,b]
             Infinite series:
               p-series and geometric series
               Absolute and conditional convergence
               Comparison, ratio, and alternating series tests
             Uniform convergence:
               Continuity of the limit function
               Integration of sequences and series
               Differentiation of sequences and series
               Weierstrass M-test
               Proving uniform convergence (or lack thereof) in specific examples
             Power series:
               Radius and interval of convergence, behavior at endpoints
               Continuity, differentiation, and integration of power series
             Definitions and statements of the basic theorems are important