Foundations of Multivariable Calculus – MATH 530

CG • Section 8WK • 11/08/2019 to 04/16/2020 • Modified 02/09/2022

Course Description

This course focuses on developing an understanding of multivariate calculus and its structure as well as how it applies to the study of functions and analytic geometry through both examples and proofs.

For information regarding prerequisites for this course, please refer to the Academic Course Catalog.


This course is an introduction to real multivariate calculus at the graduate level. After successful completion of this course the student should be able to calculate and explain all the major theorems and results of standard multivariable calculus.

Topics covered include the analysis of limits, continuity, differentiation, integration and power series for functions in one or more variables are developed. In addition, applications to optimization in several variables are discussed. The analytic geometry of curves and surfaces are analyzed with calculus.  Integration of vector fields along curves and surfaces is developed and the major theorems of vector calculus are detailed and applied. This course of study should prepare both teachers and students for further study or instruction of multivariable calculus.

Measurable Learning Outcomes

Upon successful completion of this course, the student will be able to:

  1. Define the limit using the epsilon delta formulation and prove basic results in that formalism.
  2. Define differentiability of a map in terms of limits and prove basic results in that formalism.
  3. Prove the Fundamental Theorem of Calculus.
  4. Calculate partial derivatives and multiple integrals.
  5. Optimize functions of several variables either with or without constraints.
  6. State and apply the inverse and implicit function theorems.
  7. Analyze tangent and normal spaces to manifolds in Rn.
  8. Calculate line and surface integrals.
  9. State and apply the major theorems of vector calculus.
  10. Calculate exterior derivatives and state the Generalized Stokes’ Theorem.

Course Assignment

Textbook readings and lecture presentations/notes

Course Requirements Checklist

After reading the Course Syllabus and Student Expectations, the student will complete the related checklist found in the Course Overview.

Student Introduction Video Assignment

The purpose of this video is to introduce the student and prepare the student for future video presentations.

Video Demonstrations (4)

The student will create 4 video demonstrations based on the provided prompts. Videos must be precise, well-practiced, and a maximum of 8 minutes.

Homework (6)

Homework problems are essential to this course and students will be assigned homework problems to complete throughout the course. One homework problem out of the assigned problems will be submitted for grading in Modules 1-3 and 5-7.

Homework Portfolios (2)

Homework will be assigned weekly. Homework assignments in Modules 1–4 will be scanned into a single document and submitted as Homework Portfolio: Midterm Assignment before the Midterm Exam.  Homework assignments in Modules 5–8 will be scanned into a single document and submitted as Homework Portfolio: Final Assignment before the Final Exam. Many exam problems will come from the assigned homework.

Quizzes (8)

Each quiz will be timed, handwritten, and open-book/open-notes/open-video and will cover material from the videos in the assigned modules. The time limit for each quiz is 30 minutes.  The quizzes are some combination of T/F, multiple choice and/or fill in the blank questions.

Exam Assignments (2)

Each exam will be timed, handwritten, and open-book/open-notes/open-video and will cover the course material for the assigned modules. The time limit for the Midterm Exam is 90 minutes (1 hour, 30 minutes) while the time limit for the Final Exam is 180 minutes (3 hours). On all written work, the student is expected to write correct mathematics to avoid point deductions.