# BSc Actuarial Science and Mathematics / Course details

Year of entry: 2022

## Course unit details:Calculus of Several Variables

Unit code MATH20132 10 Level 2 Semester 2 Department of Mathematics No

### Overview

Functions of several variables were briefly considered in first year calculus courses when the notion of partial derivative was introduced. Although there are some similarities with the familiar theory of one real variable, the theory for functions of several variables is far richer. For example, for functions of several variables, the critical points might be maxima, minima or saddle points (which are minima in one direction and maxima in another direction). A key idea is to generalize the definition of the derivative at apoint to the the derivative of a map  <i>f</i>:<b>R</b><sup><i>n</i></sup>&rarr;<b>R</b><sup><i>m</i></sup> at a point <i>a</i> of <b>R</b><sup><i>n</i></sup>. This is the Fr&eacute;chet derivative, which is a linear map <i>df</i>(<i>a</i>):<b>R</b><sup><i>n</i></sup>&rarr;<b>R</b><sup><i>m</i></sup> (often represented by a matrix whose entries are partial derivatives) which gives the best approximation to the function at the point <i>a</i>. This derivative is used in a number of very elegant and useful results, in particular the Inverse Function Theorem and the Implicit Function Theorem, and is a key notion in the study of the critical points of functions of several variables.

The Fr&eacute;chet derivative is an example of a differential 1-form on  <b>R</b><sup><i>n</i></sup> and so naturally leads on to an introduction to the basic ideas of differential <i>k</i>-forms. Differential <i>k</i>-forms are fundamental in the integral calculus of functions of several variables and this is briefly considered.

### Aims

The aim of this lecture course is to introduce the basic ideas of calculus of several variables.

### Learning outcomes

On the successful completion of this lecture students should:

• state the definitions of limit, directional limit and limit along a curve of a function of several variables; calculate these limits for simple examples; prove and apply the Rules for Limits to calculations for more complicated functions,
• state the definition of continuity of a function of several variables; prove that given functions are continuous for some simple examples; prove and apply the Rules for Continuous functions to more complicated functions,
• state the definitions of directional and Fr&eacute;chet derivatives; calculate these derivatives for some simple examples; prove a connection between the two derivatives; prove and apply the Rules for Derivatives to calculations for more complicated functions,
• calculate Jacobian matrices and Gradient vectors; prove relations between these and derivatives; apply these relations to calculate derivatives,
• state and apply the Chain Rule, Implicit Function Theorem and the Inverse Function Theorem,
• define and calculate the Tangent Space at a point on a surface; prove results identifying the critical points of a function restricted to a surface; apply the method of Lagrange multipliers to simple extremum problems with a constraint,
• define differential 1-form and 2-forms on an open subset of <b>R</b><sup><i>n</i></sup>; evaluate such forms at a point; evaluate the wedge product of two forms and the derivative of a form; evaluate line integrals of 1-forms and surface integrals of 2-forms over a surface parametrized by a rectangle; state a form of Stokes’ Theorem on a surface.

### Syllabus

1.Continuous functions of several variables.

2.Differentiation of real-valued functions of several variables.

3.Critical points and higher partial derivatives.

4.Differentiation of vector-valued functions of several variables.

5.Differential forms and integration of differential forms.

### Assessment methods

Method Weight
Other 20%
Written exam 80%
• A coursework test in first week after Easter (to be confirmed - Easter is very late next year so it may be before Easter): weighting 20%;
• End of semester examination: weighting 80%.

### Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

### Recommended reading

• M.J. Field, Differential Calculus and its Applications, Van Nostrand 1976.
• W. Fleming, Functions of Several Variables, Addison-Wesley 1965.
• J. and B. Hubbard, Vector Calculus, Linear Algebra, and Differential Forms, Prentice Hall 1998.
• C.H. Edwards, Jr., Advanced Calculus of Several Variables, Dover Publications 1994.
• R. Courant and F. John, Introduction to Calculus and Analysis, Volume 2, Wiley 1974.
• H.M. Edwards, Advanced Calculus: a Differential Forms Approach, Birkhauser 1994.

### Study hours

Scheduled activity hours
Lectures 12
Tutorials 12
Independent study hours
Independent study 76

### Teaching staff

Staff member Role
Mark Coleman Unit coordinator

### Additional notes

The independent study hours will normally comprise the following. During each week of the taught part of the semester:

·         You will normally have approximately 60-75 minutes of video content. Normally you would spend approximately 2-2.5 hrs per week studying this content independently

·         You will normally have exercise or problem sheets, on which you might spend approximately 1.5hrs per week

·         There may be other tasks assigned to you on Blackboard, for example short quizzes or short-answer formative exercises

·         In some weeks you may be preparing coursework or revising for mid-semester tests

Together with the timetabled classes, you should be spending approximately 6 hours per week on this course unit.

The remaining independent study time comprises revision for and taking the end-of-semester assessment.

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