MATHEMATICAL ANALYSIS

Students will:

•Be familiar with, and can apply, polar coordinates and vectors in the plane. In particular, translate between polar and rectangular coordinates and can interpret polar coordinates and vectors graphically

•Become adept at the arithmetic of complex numbers. Use the trigonometric form of complex numbers and understand that a function of a complex variable can be viewed as a function of two real variables. Know the proof of DeMoivre's theorem

•Give proofs of various formulas by using the technique of mathematical induction

•Know the statement of, and can apply, the fundamental theorem of algebra

•Be familiar with conic sections, both analytically and geometrically:

a) Take a quadratic equation in two variables; put it in standard form by completing the square and using rotations and translations, if necessary; determine what type of conic section the equation represents; and determine its geometric components (foci, asymptotes, and so forth)

b) Take a geometric description of a conic section_ for example, the locus of points whose sum of its distances from (1, 0) and (-1, 0) is 6_and derive a quadratic equation representing it

•Find the roots and poles of a rational function and can graph the function and locate its asymptotes

•Demonstrate an understanding of functions and equations defined parametrically and can graph them

•Be familiar with the notion of the limit of a sequence and the limit of a function as the independent variable approaches a number or infinity. Determine whether certain sequences converge or diverge

LINEAR ALGEBRA

Students will:

•Solve linear equations in any number of variables by using Gauss-Jordan elimination

•Interpret linear systems as coefficient matrices and the Gauss-Jordan method as row operations on the coefficient matrix

•Reduce rectangular matrices to row echelon form

•Perform addition on matrices and vectors

•Perform matrix multiplication and multiply vectors by matrices and by scalars

•Demonstrate an understanding that linear systems are inconsistent (have no solutions), have exactly one solution, or have infinitely many solutions

•Demonstrate an understanding of the geometric interpretation of vectors and vector addition (by means of parallelograms) in the plane and in three-dimensional space

•Interpret geometrically the solution sets of systems of equations. For example, the solution set of a single linear equation in two variables is interpreted as a line in the plane, and the solution set of a two-by-two system is interpreted as the intersection of a pair of lines in the plane

•Demonstrate an understanding of the notion of the inverse to a square matrix and apply that concept to solve systems of linear equations

•Compute the determinants of 2 x 2 and 3 x 3 matrices and are familiar with their geometric interpretations as the area and volume of the parallelepipeds spanned by the images under the matrices of the standard basis vectors in two-dimensional and three-dimensional spaces

•Know that a square matrix is invertible if, and only if, its determinant is nonzero. Compute the inverse to 2 x 2 and 3 x 3 matrices using row reduction methods or Cramer's rule

•Compute the scalar (dot) product of two vectors in n-dimensional space and know that perpendicular vectors have zero dot product