Difference between revisions of "Math 342: Theory of Analysis 2"
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Revision as of 15:04, 3 April 2013
Contents
Catalog Information
Title
Theory of Analysis 2.
(Credit Hours:Lecture Hours:Lab Hours)
(3:3:0)
Offered
F, W
Prerequisite
Description
Rigorous treatment of calculus of several real variables; metric spaces, geometry and topology of Euclidean space, differentiation(,?) implicity(?) function theorem, integration on sets and manifolds.
Desired Learning Outcomes
Math 342 is the multivariable sequel to Math 341. It provides a rigorous treatment of multivariable calculus. Unlike the case with Math 341, Math 342 students are not required to have had an introductory course in the subject matter, so Math 342 needs to treat both computation and theory in some depth.
Prerequisites
Math 341 provides the singlevariable results on which many of the multivariable results of Math 342 are based. Math 313 provides the tools for describing the spaces in which differentiation and integration is performed.
Minimal learning outcomes
Outlined below are topics that all successful Math 342 students should understand well. As evidence of that understanding, students should be able to demonstrate mastery of all relevant vocabulary, familiarity with common examples and counterexamples, knowledge of the content of the major theorems, understanding of the ideas in their proofs, and ability to make direct application of those results to related problems, including calculations.
 Metric Spaces
 Open and closed sets
 Limit points and sequential limits
 Compactness (sequences and open covers)
 Connectedness
 Subspaces
 Boundedness
 Completeness
 Functional limits
 Continuity (including uniform and Lipschitz)
 The Contraction Mapping Theorem
 Geometry and topology of R^{n}
 Dot product and Euclidean norm
 Components and convergence
 The BolzanoWeierstrass Theorem
 The HeineBorel Theorem
 Coordinate functions and continuity
 Algebraic continuity rules
 Differentiation of f: D ⊆ R^{m} → R^{n}
 Directional, partial, and total derivatives
 Algebraic differentiation rules
 The Chain Rule
 The Mean Value Theorem
 Higherorder derivatives
 Taylor's Theorem
 The Second Derivative Test
 Lagrange multipliers
 Newton's method
 The Inverse Function Theorem
 The Implicit Function Theorem
 Integration on ndimensional subsets of R^{n}
 Integrability of continuous functions
 Iterated integrals and Fubini's Theorem
 The Jacobian and change of variables
 Polar and spherical coordinates
 Integration on manifolds
 Line and surface integrals
 Stokes' Theorem and special cases
Textbooks
Possible textbooks for this course include (but are not limited to):
Additional topics
The theory of exterior algebra and differential forms is not a required topic, but instructors might find it useful to cover it. Instructors have the option of covering either the Riemann or (probably an abbreviated version of) the Lebesgue theory of integration.
Courses for which this course is prerequisite
Math 342 is required for Math 547 and Math 565 and is recommended for some other courses.