Unit of study_

# MATH2921: Vector Calculus and Differential Eqs (Adv)

This is the advanced version of MATH2021, with more emphasis on the underlying concepts and mathematical rigour. The vector calculus component of the course will include: parametrised curves and surfaces, vector fields, div, grad and curl, gradient fields and potential functions, lagrange multipliers line integrals, arc length, work, path-independent integrals, and conservative fields, flux across a curve, double and triple integrals, change of variable formulas, polar, cylindrical and spherical coordinates, areas, volumes and mass, flux integrals, and Green's Gauss' and Stokes' theorems. The Differential Equations half of the course will focus on ordinary and partial differential equations (ODEs and PDEs) with applications with more complexity and depth. The main topics are: second order ODEs (including inhomogeneous equations), series solutions near a regular point, higher order ODEs and systems of first order equations, matrix equations and solutions, solution methods (variation of parameters, undetermined coefficients) the Laplace and Fourier Transform, elementary Sturm-Liouville theory, an introduction to PDEs, and first methods of solutions (including separation of variables, and Fourier Series). The unit then moves to topics in solution techniques for ordinary and partial differential equations (ODEs and PDEs) with applications. It provides a more thorough 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 MATH2921 Mathematics and Statistics Academic Operations 6
 Prerequisites: ? [(MATH1921 or MATH1931 or MATH1901 or MATH1906) or (a mark of 65 or above in MATH1021 or MATH1001)] and [MATH1902 or (a mark of 65 or above in MATH1002)] and [(MATH1923 or MATH1933 or MATH1903 or MATH1907) or (a mark of 65 or above in MATH1023 or MATH1003)] None MATH2021 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 regioun, volume and surface as well as solving basic differential equations with a background in a variet of texhniques and applications of mathematical analysis
• LO2. understand the definitions, main theorems, propositions, lemmata and corollaries for multivariate calculus as well as their applications to science. Also understand and the main theorems and methods of solving elementary linear differential equations as well as the applications.
• LO3. be fluent in the computation of integrals via substitution and coordinate transform methods.
• LO4. develop an appreiciation and strong working knowledge of the theory and applications of elementry vector analysis and differential equations.
• LO5. be fluent with important examples, theorems and applications and be able to implement these in a computational software package.
• LO6. present complete and mathematically rigorous solutions for problems in vector calculus and differential equations that include appropriate justification for their reasoning.
• LO7. recognise problems in mathematics and other areas of science and engineering that are amenable to mathematical analysis, and to apply th etechniques of mathematical analysis in solving them.

## Unit outlines

Unit outlines will be available 2 weeks before the first day of teaching for the relevant session.