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Unit information: Multivariable Calculus and Complex Functions in 2020/21

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Unit name Multivariable Calculus and Complex Functions
Unit code MATH20015
Credit points 20
Level of study I/5
Teaching block(s) Teaching Block 1 (weeks 1 - 12)
Unit director Professor. Liverpool
Open unit status Not open

MATH10011 Analysis, MATH10012 ODEs, Curves and Dynamics, MATH11005 Linear Algebra and Geometry



School/department School of Mathematics
Faculty Faculty of Science


Lecturers: Tannie Liverpool and Mike Blake

Unit Description

The emphasis is on basic ideas and methods; theorems will be stated, but the emphasis is on methods rather than proofs. The first half develops an understanding of multivariable calculus including the major theorems of vector calculus. The main focus is on developing differential vector calculus, tools for changing coordinate systems and major theorems of integral calculus for functions of more than one variable. This material is fundamental to physical applied mathematics and it is also relevant to the second half of the course which introduces functions of a complex variable, with a focus on holomorphic functions. It introduces differentiation and integration of functions of one complex variable, goes through the main theorems of integration, leading up to Cauchy’s Residue theorem, and contains a final part on using theorems of complex functions to solve real-valued integrals

Relation to Other Units

This unit extends elementary calculus as seen in ODEs, Curves and Dynamics.

This unit feeds into pure and applied mathematics, such as Complex Function Theory (which develops the material) and Fluid Dynamics. Applied Partial Differential Equations and Mathematical Methods also use the material.

Intended learning outcomes

At the end of the course the student should be able to:

  • demonstrate understanding of central terms such as the derivative for multivariable functions and the main integral theorems of vector calculus
  • use vector identities in differential calculus, and differential operators in curvilinear coordinate systems
  • evaluate line, surface and volume integrals
  • use the elementary properties of holomorphic functions of a complex variable
  • find power series expansions
  • integrate holomorphic and functions with and without singularities
  • master residue calculus, and apply it to real-valued integrals.

Teaching details

The unit will be taught through a combination of

  • synchronous online and, if subsequently possible, face-to-face lectures
  • asynchronous online materials, including narrated presentations and worked examples
  • guided asynchronous independent activities such as problem sheets and/or other exercises
  • synchronous weekly group problem/example classes, workshops and/or tutorials
  • synchronous weekly group tutorials
  • synchronous weekly office hours

Assessment Details

Assessment for learning/summative assessment:

  • Weekly problem sheets will be distributed which will test the students' understanding of the material through a variety of problems ranging from elementary to difficult. Set questions will be marked promptly and returned with comments. Full solutions of all problems will be distributed.

Assessment of learning/summative assessment: 90% Timed, open-book examination 10% Coursework

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.

If you fail this unit and are required to resit, reassessment is by a written examination in the August/September Resit and Supplementary exam period.

Reading and References


  • Jerrold E. Marsden and Anthony J. Tromba, Vector Calculus, ed. 5 , W. H. Freeman and Company, 2003
  • Jerrold E. Marsden and Michael J. Hoffman, Basic Complex Analysis, ed. 3 , W. H. Freeman & Company, 1999