Calculus III


1. Shot description: The course covers polar
representation of a complex number, firstorder differential equations, secondorder
differential equations, Laplace transforms, series solutions, systems of linear equations, matrices, vectors in
two and three dimensions, linear vector spaces, and applications of linear algebra
including the simplex method, systems of linear equations, matrix
arithmetic, elementary matrices and the matrix inverse, determinants, Cramer's rule,
eigenvalues and eigenvectors, diagonalization and orthogonal diagonalization. 2. Course Objective: By the end of the course you will be expected to (1) To develop the ability to solve an ordinary differential equation of first or second order. (2) To develop the ability to model certain physical phenomenon using ordinary differential equations. (3) To develop an ability to analyze a differential equation by using numerical or graphical techniques. (4) To enhance learning by examining geometric, numerical, and algebraic aspects of each problem. (5) To acquire an understanding of the breadth of mathematics by introducing applications in a wide variety of scientific fields. (6) To enhance the ability to communicate mathematical concepts through a series of written laboratory assignments and classroom discussions. (7) To select and use technology when appropriate in problem solving. (8) To develop the process of making appropriate conjectures, finding suitable means to test those conjectures, and drawing conclusions about their validity. (9) To demonstrate knowledge of: linear transformations and their representation by matrices. (10) To know the basic algebraic processes for matrices. (11) To know the transpose of a matrix; symmetric, skewsymmetric, and orthogonal matrices. (12) To reduce a matrix to rowechelon form, linear dependence and rank. (13) To apply matrices to the solution of systems of linear equations. Gaussian elimination, LUfactorizations of a square matrix. (14) To know the inverse of a nonsingular square matrix. Determinants. Cramer's rule. (15) To compute eigenvalues and eigenvectors of a square matrix. (16) To reduce orthogonal matrices to diagonal form. (17) To apply first order differential equations to the calculation of orthogonal trajectories of a oneparameter family of plane curves. (18) To know partial derivatives and their application to the finding of maxima and minima of functions of two or more variables. 3. Homework is an important part of learning mathematics and will be assigned daily. Every assigned problem should be tried and the answer checked. It is permissible to discuss problems with other students or relatives. It is not permissible to copy another student's work. Do your best to think through the problems and understand why things work the way they do. Homework should take anywhere from 30 minutes to an hour, depending on the type of assignment. Homework will be issued in class and submitted one week later. On test days all homework associated with that test will be collected and graded. Homework should be corrected and added, as the material is better understood. In class, students follow directions: (1) Be in your seat when the bell rings; (2) Do not work on other subjects; (3) No eating, drinking, or chewing; (4) Most of the time the consequences for not following rules will be remaining after class. If it becomes more of a problem, the student will be asked to stay a short time after school for a discussion. 4. Textbook: (1) P. O’Neill, Advanced Engineering Mathematics, 4^{th} ed, Brooks/Cole (1999). (2) Courant, Richard and John, Fritz. Introduction to Calculus and Analysis, New York: SpringerVerla 