Computing integrals over polynomially defined regions and their boundaries in 2 and 3 dimensions

Michael J. Wester, Yuzita Yaacob, Stanly Steinberg

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

We use the cylindrical algebraic decomposition algorithms implemented in Mathematica to produce procedures to analytically compute integrals over polynomially defined regions and their boundaries in two and three dimensions. Using these results, we can implement the divergence theorem in three dimensions or the Green's theorems in two dimensions. These theorems are of central importance in the applications of multidimensional integration. They also provide a strong correctness test for the implementation of our results in a computer algebra system. The resulting software can solve many of the two and some of the three dimensional integration problems in vector calculus textbooks. The three dimensional results are being extended. The results in this paper are being included in an automated student assistant for vector calculus.

Original languageEnglish
Pages (from-to)79-101
Number of pages23
JournalMathematics and Computers in Simulation
Volume82
Issue number1
DOIs
Publication statusPublished - Sep 2011

Fingerprint

Vector calculus
Textbooks
Computing
Algebra
Three-dimension
Two Dimensions
Computer systems
Multidimensional Integration
Divergence theorem
Students
Green's theorem
Decomposition
Three-dimensional
Computer algebra system
Decomposition Algorithm
Mathematica
Correctness
Software
Theorem

Keywords

  • Area integral
  • Cylindrical algebraic decomposition
  • Iterated integrals
  • Line integral
  • Surface integral
  • Volume integral

ASJC Scopus subject areas

  • Modelling and Simulation
  • Numerical Analysis
  • Applied Mathematics
  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Computing integrals over polynomially defined regions and their boundaries in 2 and 3 dimensions. / Wester, Michael J.; Yaacob, Yuzita; Steinberg, Stanly.

In: Mathematics and Computers in Simulation, Vol. 82, No. 1, 09.2011, p. 79-101.

Research output: Contribution to journalArticle

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