# Math 342: Theory of Analysis 2

### Title

Theory of Analysis 2.

(3:3:0)

F, W

### 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 single-variable 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.

1. 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
2. Geometry and topology of Rn
• Dot product and Euclidean norm
• Components and convergence
• The Bolzano-Weierstrass Theorem
• The Heine-Borel Theorem
• Coordinate functions and continuity
• Algebraic continuity rules

3. Differentiation of f: DRmRn
• Directional, partial, and total derivatives
• Algebraic differentiation rules
• The Chain Rule
• The Mean Value Theorem
• Higher-order derivatives
• Taylor's Theorem
• The Second Derivative Test
• Lagrange multipliers
• Newton's method
• The Inverse Function Theorem
• The Implicit Function Theorem
4. Integration on n-dimensional subsets of Rn
• Integrability of continuous functions
• Iterated integrals and Fubini's Theorem
• The Jacobian and change of variables
• Polar and spherical coordinates
5. Integration on manifolds
• Line and surface integrals
• Stokes' Theorem and special cases

### Textbooks

Possible textbooks for this course include (but are not limited to):

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.