MA 227 - Multivariable Calculus

MA 227 - Multivariable Calculus

Review of matrix operations, Cramer’s rule, row reduction of matrices; inverse of a matrix, eigenvalues and eigenvectors; systems of linear algebraic equations; matrix methods for linear systems of differential equations, normal form, homogeneous constant coefficient systems, complex eigenvalues, nonhomogeneous systems, the matrix exponential; double and triple integrals; polar, cylindrical and spherical coordinates; surface and line integrals; integral theorems of Green, Gauss and Stokes.

Prerequisites:    MA 124, MA 116

Corequisites:    MA 221

Course Objectives

The course introduces elements of linear algebra as needed to manipulate and solve linear, constant coefficient systems of first-order ODEs.  In the second part of the course, students are introduced to the most commonly used curvilinear coordinate systems and develop skill in setting up and solving double and triple integrals.  The final part of the course introduces the theorems and techniques of vector calculus essential to mathematical physics and further study in real analysis.

Learning Outcomes

Upon completing this course, it is expected that a student will be able to do the following:

Mathematical Foundations

  1. Matrices: Carry out basic operations with matrices and vectors and use the determinant to determine the solvability of a linear system of equations.
  2. Linear Algebraic Systems: Solve a system of linear algebraic equations using matrix methods.
  3. Eigenvalues and Eigenvectors: Derive the eigenvalues and eigenvectors of a square matrix.
  4. Systems of ODEs: Solve systems of linear, constant coefficient, first order ODEs using matrix methods.
  5. Double Integrals: Formulate and evaluate iterated double integrals using rectangular or polar coordinates.
  6. Triple Integrals: Formulate and evaluate iterated triple integrals using rectangular, cylindrical, or spherical coordinates.
  7. Grad, Div, Curl: Carry out calculations in rectangular coordinates involving the differential operators gradient, divergence, and curl.
  8. Line Integrals: Formulate and evaluate line integrals.
  9. Surface Integrals: Formulate and evaluate surface integrals.
  10. Conservative Vector Fields: State and apply the relationship between conservative vector fields and path independent line integrals.
  11. Green's Theorem: State and apply Green?s Theorem for planar vector fields.
  12. Stokes' and Divergence Theorems: State and apply Stokes? Theorem and the Divergence Theorem for vector fields in R3.


R. Kent Nagle, Edward B. Saff, and Arthur David Snider, Fundamentals of Differential Equations and Boundary Value Problems, Fifth Edition, Addison Wesley.

James Stewart, Calculus, Concepts and Contexts, 4th edition, Brooks/Cole, 2008.