MATH 3334 - Advanced Multivariable and Vector Calculus - University of Houston

# MATH 3334 - Advanced Multivariable and Vector Calculus

***This is a course guideline.  Students should contact instructor for the updated information on current course syllabus, textbooks, and course content***

Prerequisites: MATH 2318 and MATH 3333, or MATH 2318 and currently enrolled in MATH 3333 with approval of instructor.

Course Description: Basic theory underlying multivariable and vector calculus, with applications. Topics include topology of n-space, derivative of a multivariable function as a linear transformation and applications, multivariable Taylor theorem, the inverse and implicit function theorems; calculus of vector fields and vector differential operators, Stokes and Gauss integral theorems, physical applications.

Recommended Text

• “Advanced Calculus: A Geometric View (Undergraduate Texts in Mathematics)” by James J. Callahan, 2010. ISBN9781441973313
• “Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach”, by John H. Hubbard and Barbara Burke Hubbard, 2015. ISBN9780971576681
• “Introduction to Vector Analysis”, by Harry F. Davis and A. David Snider, 2016. ISBN9780697160997

Upon completion of this course, the students are expected to gain an understanding of the topology of R^n; multivariable differentiation; Taylor’s theorem in several variables; implicit and inverse function theorems; algebra of vectors and vector differential operators; line, surface and volume integrals, including Green’s, Stokes’ and Divergence theorems, and curvilinear coordinates.

Brief list of topics to be covered:

• Topology of R^n: open closed sets, limits of functions and continuity, the Extreme Value Theorem on R^n.
• Multivariable Differentiation: the derivative of a function as a linear map and implications. Taylor’s Theorem in several variables, Implicit and Inverse Function Theorems.
• Multivariable Integration: double and triple integrals, Fubini’s Theorem, change of variables formula.
• Scalar and Vector Fields: Scalar fields and gradients. Introduction to vectors; scalar and vector products. Vector fields and grad, div, curl, and Del notation in rectangular, cylindrical and spherical coordinates.
• Line, surface and volume integrals, characterization of irrotational fields.
• Green’s Theorem, The Divergence Theorem and Stokes' Theorem in 3 dimensions and associated theorems. Applications.