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Unit of study_

MATH2061: Linear Mathematics and Vector Calculus

This unit starts with an investigation of linearity: linear functions, general principles relating to the solution sets of homogeneous and inhomogeneous linear equations (including differential equations), linear independence and the dimension of a linear space. The study of eigenvalues and eigenvectors, begun in junior level linear algebra, is extended and developed. The unit then moves on to topics from vector calculus, including vector-valued functions (parametrised curves and surfaces; vector fields; div, grad and curl; gradient fields and potential functions), line integrals (arc length; work; path-independent integrals and conservative fields; flux across a curve), iterated integrals (double and triple integrals; polar, cylindrical and spherical coordinates; areas, volumes and mass; Green's Theorem), flux integrals (flow through a surface; flux integrals through a surface defined by a function of two variables, though cylinders, spheres and parametrised surfaces), Gauss' Divergence Theorem and Stokes' Theorem.

Code MATH2061
Academic unit Mathematics and Statistics Academic Operations
Credit points 6
(MATH1X21 or MATH1011 or MATH1931 or MATH1X01 or MATH1906) and (MATH1014 or MATH1X02) and (MATH1X23 or MATH1933 or MATH1X03 or MATH1907)
MATH2961 or MATH2067 or MATH2021 or MATH2921 or MATH2022 or MATH2922

At the completion of this unit, you should be able to:

  • LO1. solve a system of linear equations
  • LO2. apply the subspace test in several different vector spaces
  • LO3. calculate the span of a given set of vectors in various vector spaces
  • LO4. test sets of vectors for linear independence and dependence
  • LO5. find bases of vector spaces and subspaces
  • LO6. find a polynomial of minimum degree that fits a set of points exactly
  • LO7. find bases of the fundamental subspaces of a matrix
  • LO8. test whether an n × n matrix is diagonalisable, and if it is find its diagonal form
  • LO9. apply diagonalisation to solve recurrence relations and systems of DEs
  • LO10. extended (from first year) their knowledge of vectors in two and three dimensions, and of functions of several variables
  • LO11. evaluate certain line integrals, double integrals, surface integrals and triple integrals
  • LO12. evaluate certain line integrals, double integrals, surface integrals and triple integrals
  • LO13. understand the physical and geometrical significance of these integrals
  • LO14. know how to use the important theorems of Green, Gauss and Stokes.