DEPARTMENT OF MATHEMATICS URI 
MTH 362 Section 02 Spring 2003 - Topics - 

0. REVIEW: THE FUNDAMENTAL THEOREM OF CALCULUS.

CHAPTER 1. FIRST ORDER DIFFERENTIAL EQUATIONS
1.1 Ordinary differential equation, general solution, particular solution,initial value problems 
1.2 Direction fields 
1.3 Separable differential equations, reduction to separable form 
1.5 Exact differential equations, integrating factors. 
1.6 Linear differential equations
1.9 Existence theorem, uniqueness theorem. Nonuniqueness. 

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.

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 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 22. DATA ANALYSIS. PROBABILITY THEORY.
22.1 Data: Representation, Average, Spread.
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.