DEPARTMENT OF MATHEMATICS, URI.
MTH 362
Spring 2006

REVIEW: THE FUNDAMENTAL THEOREM OF CALCULUS.

CHAPTER 22. DATA ANALYSIS. PROBABILITY THEORY.

22.2 Experiments. Outcomes. Events.
22.3 Probability.
22.4 Permutations and Combinations.
22.5 Random Variables. Probability Distributions.
22.6 Mean and Variand of a Distribution.
22.7 Binomial and Poisson Distribution.
22.8 Normal Distribution.
22.9 Distributions of Several Random Variables.

CHAPTER 12. COMPLEX NUMBERS.

12.1 Complex numbers and complex plane. Real and imaginary parts. Addition, multiplication, subtraction, division. Complex conjugate.
12.2 Polar form, modulus, argument, triangle inequality, De Moivre's formula, roots of a complex number, roots of unity.
12.6 Exponential function, periodicity, solutions of equations.

CHAPTER 6. LINEAR ALGEBRA: MATRICES, VECTORS, DETERMINANTS, LINEAR SYSTEMS OF EQUATIONS.

6.1 Matrix, vector, transpose, symmetric matrix, equality, matrix addition, scalar multiplication.
6.2 Matrix multiplication, properties, triangular matrices, Applications to production problems, Markov processes.
6.3 Linear Systems of equations, homogeneous system, solution vector, trivial solution. Matrix form of the system, geometric interpretation in the plane. Gaussian elimination. Elementary row operations. Systems with infinitely many solutions, unique solution, or no solutions. Echelon form. Pivoting.
6.4 Linear combinations of vectors. Linear independence and dependence. Rank of a matrix. Vector space, dimension, basis, determining rank using gauss elimination.
6.5 Fundamental theorem for linear systems. Null space of a matrix. Homogeneous systems with fewer equations than unknowns. Characterization of solutions of a system of equations.
6.6 Determinants, minors, cofactors, properties of determinants, rank in terms of determinants, Cramer's rule.
6.7 Nonsingular and singular square matrices. Inverse of a square matrix. Existence of the inverse and rank. Finding the inverse by Gauss-Jordan elimination and by cofactors. Inverse of diagonal, triangular, and block diagonal matrices. Matrix equations. Determinant of matrix products.

CHAPTER 1. FIRST ORDER DIFFERENTIAL EQUATIONS

1.1 Ordinary differential equation, general solution, particular solution,initial value problems
1.2 Direction fields isoclines
1.3 Separable differential equations, reduction to separable form
1.5 Exact differential equations, integrating factors.
1.6 Linear differential equations, Bernoulli equation.
1.9 Existence theorem, uniqueness theorem. Nonuniqueness. Picard iteration.

CHAPTER 2. LINEAR DIFFERENTIAL EQUATIONS OF SECOND AND HIGHER ORDER

2.1 Linear differential equations of second order. Superposition principle. Fundamental theorem. Initial value problem. General solution, basis, particular solution. Method of reduction of order.
2.2 Second order homogeneous equations with constant coefficients: real roots cases.
2.3 Second order homogeneous equations with constant coefficients: complex roots case.
2.4 Differential operators. Characteristic equation.
2.5 Mass-spring systems. Undamped and damped systems.Overdamping, critical damping, underdamping.
2.8 Nonhomogeneous linear second order systems
2.9 Solving nonhomogeneous linear second order equations with constant coefficients by the method of undetermined coefficients.
2.11 Forced oscillations, Undamped and damped cases.