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Calculus 2 (MATH 20900)

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Administrative Information

  1. Unit number and title: MATH 20900 Calculus 2
  2. Level: I/5
  3. Credit point value: 20 credit points
  4. Year: 12/13
  5. First Given in this form: 1995-6. Modifications in 2010-13.
  6. Unit Organiser: Jens Eggers
  7. Lecturer: Jens Eggers and Karoline Wiesner
  8. Teaching block: 1
  9. Prerequisites: MATH 11005 Linear Algebra and Geometry, MATH 11006 Analysis 1 and MATH 11007 Calculus 1

Unit aims

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

General Description of the Unit

This unit extends elementary calculus in two ways: first to vector-valued functions of several (real) variables, and then to functions of a complex variable. The emphasis is on basic ideas and methods; theorems will be stated rigorously and the theory will be carefully developed, but the style is closer to first year calculus than to analysis.

The first half of the course develops multivariable calculus to the point where the major theorems can be presented: Green's theorem, Stokes' theorem and the divergence theorem. The second part of the course develops the basic ideas of the theory of 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.

Teaching Methods

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

Learning Objectives

At the end of the course the student should:

  • understand the definition of the derivative for multivariable functions
  • be comfortable with vector identities in differential calculus, and differential operators in curvilinear coordinate systems
  • 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.

Assessment Methods

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.

Award of Credit Points

Credit points are normally gained by passing the unit, i.e. achieving an assessment mark of at least 40.

Transferable Skills

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

Texts

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

Syllabus

Multivariable calculus:

  • Differential calculus in R^n: Matrix norm.  Summation convention. Continuity.  Differentiability. Relation to partial derivatives.  Equality of mixed partials.  Higher-order derivatives.  Taylor's theorem.
  • Differential vector calculus:  Grad, div, curl.  Identities. Levi-Cevita symbol. Differential operators in curvilinear coordinate systems.  Scale factors.
  • 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 one complex variable. Continuity. Analyticity.  Cauchy-Riemann equations. Holomorphic functions.
  • Integral Calculus. Cauchy's theorem and formulae.  Liouville's theorem.
  • Power series. Taylor's Theorem. Laurent's Threorem.
  • Residues. Isolated singularities. Residue Theorem.

Unit Home Page

Course web pages: 

For Multivariable Calculus, see http://www.maths.bristol.ac.uk/~majge/calculus2.html

For Functions of a Complex Variable, see Blackboard.