University at Buffalo GIFTED MATH PROGRAM 2009-10 Course VI Curriculum Fall Semester Course Number: MTH 241 Course Title: College Calculus 3 Credit Hours: 4.0 Textbook: Stewart, Calculus: Multivariable (Early Transcendentals, 6th ed. / UB custom 6th ed.), Brooks/Cole Description: Geometry and vectors of n-dimensional space; Green’s Theorem, Stokes’ Theorem, multidimensional differentiation and integration; application to two and three-dimensional space. Prerequisite: MTH 142 with recommended grade of “C” or higher Syllabus: MTH 241 covers Chapters 12 through Chapter 16 of the text. Week Section Topics 1 12.1 – 12.4 Three-Dimensional Coordinate Systems, Vectors, Dot Product, Cross Product 2 12.5 – 12.7 Equations of Lines and Planes, Cylinders and Quadratic Surfaces, Cylindrical and Spherical Coordinates 3 13.1 – 13.3 Vector Functions and Space Curves, Derivatives and Integrals of Vector Functions, Arc Length and Curvature 4 13.4, 14.1 Motion in Space: Velocity and Acceleration; Functions of Several Variables 5 14.2 – 14.4 Limits and Continuity, Partial Derivatives, Tangent Planes and Linear Approximation 6 14.5 – 14.7 Chain Rule, Directional Derivatives and Gradient Vector, Maximum and Minimum Values 7 14,8, 15.1-15.2 Lagrange Multiplier, Double Integrals over Rectangles, Iterated Integrals 8 15.3 – 15.5 Double Integrals over General Regions, Double Integrals in Polar Coordinates, Applications of Double Integrals 9 15.6 – 15.8 Surface Area, Triple Integrals, Triple Integrals in Cylindrical and Spherical Coordinates Option: Section 15.9 Change of Variables in Multiple Integrals 10 16.1 – 16.4 Vector Fields, Line Integrals, Fundamental Theorem for Line Integrals, Green’s Theorem 11 16.5 – 16.7 Curl and Divergence, Parametric Surfaces and their Area, Surface Integrals 12 16.8 – 16.9 Stokes’ Theorem, Divergence TheoremUniversity at Buffalo GIFTED MATH PROGRAM 2009-10 Course VI Curriculum Spring Semester Course Number: MTH 309 Course Title: Introduction to Linear Algebra Credit Hours: 4.0 Textbook(s): David Lay, Linear Algebra and its Applications, 3rd ed., Addison Wesley (UB custom edition is identical to standard editions.) Description: Linear equations, linear transformations, matrices, determinants, vector spaces, eigenvalues and eigenvectors, inner products, orthogonality, quadratic forms. Prerequisite: MTH 142 (Calculus 2) or MTH 192 (Discrete Math 2) Syllabus: Chapters 1 through 7 as specified below. Section Title Topics 1.1 – 1.8 Linear Equations in Linear Algebra Systems of linear equations. Row reduction and echelon forms. Vector equations. Ax-b. Solution sets of linear systems. Applications of linear systems. Linear independence. Linear transformations. 2.1 – 2.3 2.8 – 2.9 Matrix Algebra Matrix operations. Inverse of a matrix. Characterizations of invertible matrices. Subspaces of Rn. Dimension and rank. 3.1 – 3.2 Determinants Introduction to determinants. Properties of determinants. 4.1 – 4.6 Vector Spaces Vector spaces and subspaces. Null spaces, column spaces, and linear transformations. Linearly independent sets, bases. Coordinate systems. Dimension of a vector space. Rank. 5.1 – 5.5 Eigenvalues and Eigenvectors Eigenvectors and eigenvalues. Characteristic equation. Diagonalization. Eigenvectors and linear transformations. Complex eigenvalues. 6.1 – 6.5 Orthogonality and Least Squares Inner product, length, orthogonality. Orthogonal sets. Orthogonal projections. Gram-Schmidt process. Least squares problem. 7.1 – 7.2 Symmetric Matrices and Quadratic Forms Diagonalization of symmetric matrices. Quadric
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