Unit name | Calculus 2 |
---|---|
Unit code | MATH20900 |
Credit points | 20 |
Level of study | I/5 |
Teaching block(s) |
Teaching Block 1 (weeks 1 - 12) |
Unit director | Professor. Robbins |
Open unit status | Not open |
Pre-requisites |
MATH11002 and MATH11003 |
Co-requisites |
None |
School/department | School of Mathematics |
Faculty | Faculty of Science |
This unit extends elementary calculus in two ways: first to the calculus of several variables, and then to the case where the variable is a complex number. The emphasis will be on basic ideas and methods; theorems will be stated rigorously, and attention will be paid to the logical structure of the theory, but the style will be closer to first year calculus than to analysis. The first half develops multivariable calculus to the point where the major theorems can be given: the divergence theorem, and Green's and Stokes's theorems. This material is fundamental to physical applied mathematics; and it is also relevant to the second half of the course. The second half introduces the basic ideas of functions of a complex variable, including differentiation and integration in the complex plane. It includes techniques useful in applied mathematics as well as theoretical concepts.
Aims:
Syllabus
Multivariable calculus:
Differential calculus in R^n: Matrix norm. Continuity. Differentiability. Relation to partial derivatives. Equality of mixed partials. Higher-order derivatives. Taylor's theorem. Criteria for local minima/maxima.
Functions of a complex variable:
Relation to Other Units This unit is central to a good deal of pure and applied mathematics. MATH 20402 Applied Partial Differential Equations and MATH 30800 Mathematical Methods use the material of Calculus 2. MATH 32900 Differentiable Manifolds and MATH 33000 Complex Function Theory develop the multivariable calculus and complex variables material, respectively. The first half of Calculus 2 is available as Multivariable Calculus MATH 20901.
At the end of the course the student should:
Transferable Skills:
Clear logical thinking, problem solving, assimilation of abstract ideas and application to particular problems.
Lectures (33 in all), problems classes, homework and solutions (issued later).
The final mark for Calculus 2 is calculated from a 2 ½ -hour written examination in April consisting of SIX questions. The questions are divided into two groups of three questions based on the two halves of the unit, namely multivariable calculus and functions of a complex variable. A candidate's best TWO answers from each group for a total of FOUR answers will be used for assessment. Calculators are NOT permitted.
Multivariable calculus: Jerrold E. Marsden & Anthony J. Tromba, Vector Calculus, ed. 5 , W. H. Freeman and Company, 2003
Functions of a complex variable: Jerrold E. Marsden & Michael J. Hoffman, Basic Complex Analysis, ed. 3 , W. H. Freeman & Company, 1999.