Multivariable and Complex Calculus
Go to this course in the University Course Planner.
Description
The mathematics required to describe most "real life" systems involves functions of more than one variable, so the differential and integral calculus developed in a first course in Calculus must be extended to functions of more variables. In this course, the key results of one-variable calculus are extended to higher dimensions: differentiation, integration, and the link between them provided by the Fundamental Theorem of Calculus are all generalised. The machinery developed can be applied to another generalisation of one-variable Calculus, namely to complex calculus, and the course also provides an introduction to this subject. The material covered in this course forms the basis for mathematical analysis and application across an extremely broad range of areas, essential for anyone studying the hard sciences, engineering, or mathematical economics/finance.
Objective
The course is intended to equip students with some of the notions,
techniques and results required in many specialised areas of mathematics
and physics that they may go on to study, including geometry, topology,
optimisation, differential equations, electromagnetism, and solid and
fluid mechanics.
Content
Topics covered are: introduction to multivariable calculus; differentiation of scalar- and vector-valued functions; higher-order derivatives, extrema, Lagrange multipliers and the implicit function theorem; integration over regions, volumes, paths and surfaces; Green's, Stokes' and Gauss's theorems; differential forms; curvilinear coordinates; an introduction to complex numbers and functions; complex differentiation; complex integration and Cauchy's theorems; and conformal mappings.
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| Year |
Semester |
Level |
Units |
| 2012 |
1 |
2 |
3 |
Delivery
42 hours of lectures and tutorials.
Assessment
Coursework assessment 15%, mid-semester test 15%, final exam 70%.
Graduate attributes
Linkage past
Students should have successfully completed the course MATHS 1012 Mathematics IB, MATHS 2004 Mathematics IIM, or equivalent.
Linkage present
The techniques developed in this course may be used to derive many of the equations studied in MATHS 2102 Differential Equations.
Linkage future
The course provides an essential foundation for many of the Level III courses in the disciplines of Pure and Applied Mathematics.
Restrictions
May not be presented with PURE MTH 2005, PURE MTH 3016, MATHS 2202 or Real Analysis prior to 2002. May not be presented with VACA.
Recommended text
Vector Calculus, J. E. Marsden and A. J. Tromba (Barr Smith Library 517 M364v.5)
Vector Calculus, P. C. Matthews (Barr Smith Library 514.7424 M441v)
Basic Complex Analysis, J. E. Marsden and M. J. Hoffman (Barr Smith Library 517.54 M364b)
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