7/19/23, 1:24 PM Course Syllabus
College of Science and Engineering - Department of Mathematics
Designator Course Number Course Title Credit Hours/Mode
MTH 533 Advanced Calculus II 3(3-0)
Bulletin Description:
Continuation of MTH 532. Rigorous development of calculus of several variables. Limits, continuity,
differentiation, integration, implicit and inverse function theorems, differential forms. Prerequisite: MTH
532.
Course Modality: Face-to-Face
Prerequisites: MTH 532
Pre/Co-requisites: None.
Co-requisites: None.
Recommended: None.
Rationale for Course Level: The amount and complexity of mathematical topics such as topological
properties of Euclidean spaces, differentiability properties of maps, integration of differential forms
covered in this course makes it appropriate only for graduate or advanced undergraduate students. From
the prerequisite course MTH 532, we need in this course many topics such as continuity, differentiation in
one variable, integration in one variable etc. The main topics in this course are differentiation and
integration of functions of several variables and the application of these concepts.
Suggested Textbooks:
Conway, John B. , A first course in Analysis, Cambridge Mathematical Textbooks, 2017
or
Wade, William R. , An Introduction to Analysis, Prentice-Hall, fourth edition, 2010 (classic text)
Other Requirements and/or Materials for the Course:
None.
Student Learning Outcomes:
After successful completion of this course, the student will be able to:
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7/19/23, 1:24 PM Course Syllabus
1. State and prove the basic properties of functions of several real variables.
2. Compute the derivatives of maps between Euclidean spaces, their Jacobians, and integrals of
functions and differential forms of several variables.
3. Compute volumes of regions in Euclidean spaces and surface measures of hypersurfaces and other
submanifolds in Euclidean spaces.
4. Use standard results such as change of variables formula, Stokes’ formula, implicit and inverse
function theorems to compute quantities related to and prove facts about functions of several
variables.
Suggested Course Outline:
Metric and Euclidean Spaces (3 Weeks)
Sequences, completeness, continuity
Open, closed, compact and connected sets
The space of continuous functions in the sup norm
Differentiation (3 Weeks)
Differentiability in several variables, Jacobian matrix
Inverse and implicit function theorems
Critical points and Lagrange multipliers
Integration (3 Weeks)
Riemann integration in several variables
Computing integrals using iterated integration and change of variables
Differentiation under the integral sign
Fundamental theorem of Calculus in two and three dimensions (2 Weeks)
Curves and surfaces in two and three dimensions
Green’s theorem
Divergence and Stokes’ theorem in three dimensions
Differential Forms (4 Weeks)
Definition of differential forms in Euclidean space
Basic operations on forms
Integration of forms on Chains
Stokes’ formula for forms
Introduction to de Rham cohomology
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7/19/23, 1:24 PM Course Syllabus
Final Examination (1 Week)
Suggested Evaluation:
Homework assignments 20%
Two midterms 50%
Final exam 30%
Graduate students will be assigned additional and more challenging homework problems. Grades will be
computed using the same scale.
Syllabus Prepared By: Debraj Chakrabarti, Ph. D.
Date: 11/21/2018
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