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Multivariable Calculus and Mathematica

Jonathan Rosenberg
Organization: University of Maryland
Department: Mathematics
Education level


The course aims to teach the principles of multivariable calculus in a modern way, using Mathematica. We introduce students to aspects of elementary differential geometry, optimization and physics that, while important and relevant to the needs of practicing scientists and engineers, are often omitted in a traditional text. The emphasis is on the geometric, symbolic, numeric, and qualitative aspects of the subject. The problems are designed to force the student to engage in critical, analytic, and interpretive thinking beyond rote manipulation of algebra and calculus formulas.

The course meets 3 hours a week in a traditional classroom, and one hour per week in a computer lab where students work on problems in an interactive setting. We use three different kinds of course materials:
  • Calculus with Analytic Geometry, 5th ed. by Ellis and Gulick, Chs. 11-15. This is a traditional multivariable calculus textbook, although almost any other calculus book could be substituted.
  • Multivariable Calculus and Mathematica by Coombes, Lipsman, and Rosenberg (TELOS, Springer, 1998).
  • Approximately 14 Mathematica notebooks (only partially completed) that we have prepared for the lab sessions.
Each session begins with a warm-up problem, a tutorial, and problems for the students to work on by themsleves or in small groups, and concludes with a quiz problem. The most unusual feature of the course is the way we handle homework and exams. Homework problems are assigned primarily from the C-L-R text, though sometimes routine problems are added from the traditional text. The students work on these in Mathematica and submit their solution notebooks by ftp. The exams are conducted online, in Mathematica, in a computer lab.

This course is the third semester of the caculus sequence for science and engineering majors. Most of the students are sophomores, however a few are freshmen with advanced placement.

This format for multivariable calculus is not necessarily for all students, but has been quite successful with engineering and science students interested in modern computing. Many students come out of this course not only knowing the material thoroughly, but also sufficiently adept at Mathematica to use it in their other science and math courses.

  • Vectors, lines and planes
  • Geometry of curves, with applications to kinematics
  • Partial and directional derivatives
  • Optimization in several variables
  • Multiple integrals, including change of variables
  • Calculus of vector fields, including Green's Theorem, Stokes' Theorem, and the Divergence Theorem

*Applied Mathematics > Optimization
*Mathematics > Calculus and Analysis > Calculus
*Mathematics > Geometry > Computational Geometry