Advanced College Courses

Covers foundational topics in math starting with definitions of sets, functions and relations, as well as logic and methods of proof, such as proof by induction.  Also includes discrete probability and counting via permutations and combinations.  Additional topics can include graphs and trees.  Discussions of algorithm efficiency can also be available on request. 

Reviews vectors in two and three dimensions, with brief discussion of vectors in R^n, as well as in-depth coverage of parametric equations.  Explores extending functions of a single variable via   vector valued functions and multivariable functions.  Can also include discussions of Fourier and Laplace transforms.

Topics include multiple integrals, line integrals and vector fields. Culminates with in-depth coverage of the theorems of Gauss, Green and Stokes.

Includes the usual topics of systems of equations and Gaussian elimination.  Matrices are covered in detail, which includes matrix addition, multiplication, finding inverses and determinants.  Eigenvalues and eigenvectors are covered in detail.    

n-dimensional vector spaces and algebra in R^n, dot and cross products (in 3-d), linear independence and orthogonality.  Linear transformations between vector spaces represented by matrices and properties of linear mappings, e.g., being 1-1 (one-to-one), onto & isomorphic.  Also covers images and kernels of linear mappings.   Spectral theory can also be requested.  

Basic ideas of order and degree for differential equations.  First order differential equations, their method of solution and applications.  Second order differential equations, their method of solution along with extensive applications to physics and engineering.  Laplace transforms as a method to solve ODEs are also addressed.

Topics are focused on applications to solve real-world problems.  Distinguishes between initial conditions and boundary value problems.  Covers the Method of Characteristics for first order PDEs.  Higher order PDEs are solved by a range of methods, including separation of variables, expansion in eigenfunctions and integral transforms.

More in-depth than the AP / College introductory subject listed above.  Expanded coverage of discrete and continuous probability distributions and their interrelations.  Addresses foundation of statistical methods and data analysis techniques.  Again, numerous examples!

Complex variables underlie much of mathematics.  Topics include algebra of complex numbers, Argand diagrams, Cauchy-Riemann equations, Cauchy's integral formula, Laurent series, contour integration, method of residues, conformal mapping and analytic continuation.