Vector Calculus

Course #MA1116

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Clock Est. completion in 5 Weeks

Location pin Offered through Distance Learning

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This course reviews the calculus of vector fields and some topics from linear algebra.

Vector calculus topics include directional derivative, gradient, divergence, curl; potential fields; Green’s, Stokes’, and Gauss’ Divergence theorems, applications in engineering and physics.
Matrix algebra topics covered are: the fundamental algebra of matrices including addition, multiplication of matrices, multiplication of a matrix by a constant and a column (vector) by a matrix; elementary matrices and inverses, together with the properties of these operations; solutions to mxn systems oflinear algebraic equations using Gaussian elimination and the LU decomposition (without pivoting); determinants, properties of determinants; Taught at the rate of nine hours per week for five weeks.

Prerequisites

  • MA1115

Learning Outcomes


Linear Algebra

  • Use Gauss-Jordan elimination to find the general solution of a linear system with m equations and n unknowns, and determine the type of solution set.
  • Perform algebraic operations on matrices and vectors: addition, subtraction, scalar multiplication, matrix multiplication and transposition.
  • Apply the basic properties of the inverse of a matrix to simplify matrix expressions (and solve linear systems) and find the inverse of a square matrix using the Gauss-Jordan method.
  • Compute the determinant of a square matrix either by elementary row operations (EROs) or by co-factor expansion and use determinants to solve systems (Cramer's rule) and find the inverse of a matrix.
  • Find the eigenvalues and associated eigenvectors of square matrices, including cases of repeated or complex eigenvalues.
  • Model and analyze discrete dynamic systems by using eigenvalues and eigenvectors to find the steady-state vector and predict the long-term behavior of a Markov chain.

Vector Calculus

  • Define the concept of a vector field (with examples), calculate and interpret the curl and divergence of a vector field.
  • Classify vector fields as conservative or nonconservative and find the scalar potential function of a conservative vector field.
  • State and explain the generalizations of the Fundamental Theorem of Calculus and apply the theorems of vector calculus, such as the Fundamental Theorem of Line Integrals, Green's Theorem, the Divergence Theorem, and Stokes' Theorem, to simplify integration problems.
  • Apply the computational and conceptual principles of vector calculus to solve engineering and physics problems.