Unit information: Calculus 2 in 2014/15

Unit name	Calculus 2
Unit code	MATH20900
Credit points	20
Level of study	I/5
Teaching block(s)	Teaching Block 4 (weeks 1-24)
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

Description including Unit Aims

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:

• To develop an understanding of multivariable calculus including the major theorems of vector calculus.
• To introduce functions of a complex variable, especially holomorphic functions.
• To show connections between the theories of two-dimensional vector fields and functions of a complex variable.

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.

• Differential vector calculus: Grad, div, curl. Identities. Levi-Cevita symbol (from 2011).
• Integration in vector calculus. Line integrals of vector fields. Surface integrals. Stokes' theorem. Three-dimensional integrals. Gauss' theorem.

Functions of a complex variable:

• Functions of a complex variable: continuity, analyticity, the Cauchy-Riemann equations, harmonic functions.
• Holomorphic functions. Cauchy's theorem and formula. Differentiable implies infinitely differentiable. Liouville's theorem. Taylor Series.
• Laurent series. Isolated singularities. Residues.

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.

Intended Learning Outcomes

At the end of the course the student should:

• understand the definition of the derivative for multivariable functions
• be able to calculate Taylor series and identify local maxima/minima for multivariable functions
• understand and be able to evaluate line, surface and volume integrals
• understand the main integral theorems of vector calculus
• be familiar with and be able to use the elementary properties of analytic functions of a complex variable.
• be able to recognise isolated singularities of functions of a complex variable and evaluate residues.

Transferable Skills:

Clear logical thinking, problem solving, assimilation of abstract ideas and application to particular problems.

Teaching Information

Lectures (33 in all), problems classes, homework and solutions (issued later).

Assessment Information

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.

Reading and References

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.