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Unit of study_

MATH2021: Vector Calculus and Differential Equations

This unit opens with 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, through cylinders, spheres and other parametrised surfaces), Gauss' and Stokes' theorems. The unit then moves to topics in solution techniques for ordinary and partial differential equations (ODEs and PDEs) with applications. It provides a basic grounding in these techniques to enable students to build on the concepts in their subsequent courses. The main topics are: second order ODEs (including inhomogeneous equations), higher order ODEs and systems of first order equations, solution methods (variation of parameters, undetermined coefficients) the Laplace and Fourier Transform, an introduction to PDEs, and first methods of solutions (including separation of variables, and Fourier Series).

Code MATH2021
Academic unit Mathematics and Statistics Academic Operations
Credit points 6
(MATH1X21 or MATH1931 or MATH1X01 or MATH1906) and (MATH1XX2 or a mark of 65 or above in MATH1014) and (MATH1X23 or MATH1933 or MATH1X03 or MATH1907)
MATH2921 or MATH2065 or MATH2965 or (MATH2061 and MATH2022) or (MATH2061 and MATH2922) or (MATH2961 and MATH2022) or (MATH2961 and MATH2922) or MATH2067

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

  • LO1. demonstrate a conceptual understanding of vector-valued functions, partial derivatives, curves, and integration over a region, volume, and surface as well as solving basic differential equations thorough background in a variety of techniques and applications of mathematical analysis.
  • LO2. understanding the definitions, main theorems, and corollaries for path integrals, conservative fields, multiple integrals Green's theorem, Gauss' theorem, and Stokes' theorem, but also to understand the structure of the solutions of linear differential equations, the method of series solutions, the Laplace transform, solving boundary-value problems involving Sturm-Liouville operators on either a bounded interval or a rectangle, and to understand what is an eigenvalue.
  • LO3. be fluent with substitutions in integrals and changing coordinate systems from cartesian into polar, cylindrical, or spherical ones.
  • LO4. develop appreciation and strong working knowledge of the theory and applications of elementary Vector Analysis and Differential Equations.
  • LO5. be fluent with important examples, theorems, and applications of the theory.