Topics include: include: linear systems, matrix algebra, Rn as a vector space, a normed space and an inner-product space, linear transformations on Rn, eigenvalues, applications to circuits, mechanics and an introduction to computer methods.

Topics include: limits and continuity; differentiation; applications of the derivative - related rates problems, curve sketching, optimization problems, L'Hopital's rule; definite and indefinite integrals; the Fundamental Theorem of Calculus; applications of integration in geometry, mechanics and other engineering problems.

Topics include: techniques of integration, an introduction to mathematical modeling with differential equations, infinite sequences and series, Taylor series, parametric and polar curves, and application to mechanics and other engineering problems.

This course covers systems of linear equations and Gaussian elimination, applications; vectors in Rn, independent sets and spanning sets; linear transformations, matrices, inverses; subspaces in Rn, basis and dimension; determinants; eigenvalues and diagonalization; systems of differential equations; dot products and orthogonal sets in Rn; projections and the Gram-Schmidt process; diagonalizing symmetric matrices; least squares approximation. Includes an introduction to numeric computation in a weekly laboratory.

Topics include: limits and continuity, differentiation, maximum and minimum problems, definite and indefinite integrals, application of integration to geometry, mechanics and other engineering problems as well as an introduction to first order differential equations.

Ordinary differential equations. Classification. Equations of first order and first degree. Linear equations of order n. Equations of second order. Bessel's equation. Legendre's equation. Series solutions. Systems of simultaneous equations. Partial differential equations. Classification of types. The diffusion equation. Laplace's equation. The wave equation. Solution by separation of variables.

Ordinary differential equations. Equations of first order and first degree. Linear equations of order n. Systems of simultaneous equations. Difference equations. Forecasting. Business dynamics. Basic Set Theory. Counting, Cartesian Product, Combinations, Permutations. Basic Propositional Logic and Proofs. Throughout the course: formulating and analysing differential equation, difference equation, and discrete mathematical models for real-world problems.

An introduction to complex variables and ordinary differential equations. Topics include: Laplace transforms, ordinary higher-order linear differential equations with constant coefficients; transform methods; complex numbers and the complex plane; complex functions; limits and continuity; derivatives and integrals; analytic functions and the Cauchy-Riemann equations; power series as analytic functions; the logarithmic and exponential functions; Cauchy's integral theorem, Laurent series, residues, Cauchy's integral formula, the Laplace transform as an analytic function. Examples are drawn from electrical systems.

The chain rule for functions of several variables; the gradient, directional derivative, tangent plane and small signal modeling and Jacobians. Multiple integrals; change of variables and Jacobians, line integrals: parametric and explicit representations, the divergence and curl of a vector field, surface integrals; parametric and explicit representations, multi-variable Dirac Delta distribution, superposition of vector fields, Helmholtz decomposition theorem, Divergence theorem and Stokes' theorem and application from electromagnetic fields.

Existence and uniqueness of solution for first-order differential equations, general second-order linear ODEs, homogeneous equations, nonhomogeneous equations, variable coefficients, variation of parameters, Systems of ODEs, Fourier series, Laplace transform, interpretation of problems in mathematical terms, single-step numerical methods for ODEs. Introduction to Partial Differential Equations.

Partial differentiation, grad, div, curl, multiple integrals, line integrals, surface integrals, differential equations, first order differential equations, homogeneous linear differential equations, boundary conditions. Formulation of various problems relevant to materials and mining engineering - the concepts above are used.

This course provides the foundations of analysis and rigorous calculus for students who will take subsequent courses where these mathematical concepts are central of applications, but who have only taken courses with limited proofs. Topics include topology of Rn, implicit and inverse function theorems and rigorous integration theory.

Construction of Real Numbers. Metric spaces; compactness and connectedness. Sequences and series of functions, power series; modes of convergence. Interchange of limiting processes; differentiation of integrals. Function spaces; Weierstrass approximation; Fourier series. Contraction mappings; existence and uniqueness of solutions of ordinary differential equations. Countability; Cantor set; Hausdorff dimension.

Function spaces; Arzela-Ascoli theorem, Weierstrass approximation theorem, Fourier series. Introduction to Banach and Hilbert spaces; contraction mapping principle, fundamental existence and uniqueness theorem for ordinary differential equations. Lebesgue integral; convergence theorems, comparison with Riemann integral, L^p spaces. Applications to probability.

Course examines the following: analytic functions, Cauchy-Reimann equations, contour integration, Cauchy's theorem, Taylor and Laurent series, singularities, residue calculus, conformal mapping, harmonic functions, Dirichlet and Neumann problems and Poisson integral formulas. Course includes studies of linear differential equations in the complex plane, including Bessel and Legendre functions.