The official source of material for this course is the syllabus page, this page will be a source of material for the course.

To give you a brief idea of what you can expect to be able to do at the end of the course here are the Intended Learning Outcomes:

• 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 Fréchet 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 k-forms on an open subset of Rn; 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.

## Feedback on the 2017-18 Exam

Exam with Solutions
Feedback

## The Timetable 2017-18

• 2 Lectures per week
• 1 example classes per week per student.
9.00 10.00 11.00 12.00 1.00 2.00 3.00 4.00 5.00
Monday
Feedback Tutorial
Zochonis B26

Tuesday     Lecture
Stopford
Th 1
Feedback Tutorial
Samuel Alexander
A116

Wednesday
Thursday   Feedback Tutorial
Samuel Alexander A102
Lecture
Stopford
Th 1

Friday

## Past & Present exam papers

Exam Papers Solutions and
Feedback
June 2015 2015 Solutions
June 2016 2016 Solutions 2016 Feedback
June 2017 2017 solutions & Feedback

## Lecture Notes

Notes Contents
1. Continuity Functions of Several Variables: Notation; Limits of vector-valued functions; Scalar-valued examples; Sandwich Rule; Vector-valued examples; Limit Laws; Directional limits; Limits along curves; Continuity of vector-valued functions; Continuity Laws; Composition Laws. [4]
1. Appendix Appendix Uniqueness of Limit; Example of Limit; Directional limits; Composition Results.
2. Differentiation 1

Differentiation of functions of several variables: Directional derivatives; Partial Derivatives; Differentiable implies Continuous; Vector-valued Linear Functions; Fréchet Derivative.

2. Differentiation 2

Fréchet Differentiable implies Continuous; Derivative exists implies directional derivative exists; Jacobian Matrix; Special cases: functions of one variable, Gradient vector; Use of the Jacobian matrix; When is a function differentiable? C1 functions are Fréchet differentiable.

2. Differentiation 3

Examples; Product and Quotient functions; The Chain Rule; Rules for differentiation; Special cases; the Inverse function.

2. Appendix Appendix Partial and directional derivatives; Directional derivative implies directional continuity; Derivative is unique; Mean Value Theorem; Differentiable does not implies C1; Earlier example revisited; Rules of Differentiation; Chain Rule.
3. Surfaces 1

Surfaces and the Implicit Function Theorem: Surfaces; a graph as a surface; An Image set as a surface: parametric description; A Level set as a surface: Implicit description;

3. Surfaces 2

Level sets are graphs (locally): Implicit Function Theorem; Parametric sets are graphs (locally): Inverse Function Theorem;

3. Surfaces 3 Manifolds. Not examinable.
3. Surfaces 4 Planes; Tangent Spaces and Tangent Planes; Tangent Space for a graph; Tangent Space for an Parametric set; Tangent Space for a Level Set;
3. Conclusion
3. Appendix Appendix on Surfaces Full Rank; Jacobian matrix of a graph written as an image set; Jacobian matrix of a graph written as a level set; Surface of a unit ball in R3; Why did I define a surface as a graph in the way I did?; Permuting the coordinate functions in a parametric set; permuting the variables in a level set; Tangent Space for a Parametric set; Surfaces as level sets f-1(0). Which f?; Definition of Tangent Plane; Background Linear Algebra; Tangent Space for a Parametric Set; Tangent Space for a Level Set; Proof of the Inverse Function Theorem.
3. Tangent Spaces Appendix on Tangent Spaces Tangent Spaces for a surface in R3 with illustrations.
4. Implicit Function Theorem Proof of the Implicit Function Theorem: by Induction. No time in lectures & not examinable.
5. ExtremAl Values Extremal Values: Extremal Values and Lagrange multipliers;
5. Appendix Appendix Examples including the GM-AM inequality & the Cauchy-Schwarz inequality along with their extensions.
6. Forms and Integration Differential forms and their integration: Differential 1-forms; Exact 1-forms; higher-order derivatives; Closed 1-forms; line integrals; Differential 2-forms; Surface integrals; Products of 1-forms; Derivatives of 1-forms; Stoke's Theorem (statement of); Green's Theorem (statement of); These later topics will only be covered if there is time: Differentiable k-forms; products and derivatives of k-forms; integration of a k-form over a manifold.
6. Appendix Appendix Projections are linearly independent; Second derivatives need not be equal; motivation for the definition of a 2-form; Proof of Stoke's Theorem; Vector fields; A basis for k-forms.

Here are some additional notes I have typed up. They might help you as background, but they are not examinable.

Notes Contents
Higher Order Derivatives Higher Order Derivatives: Higher order partial derivatives; The Hessian Matrix.
Taylor Series Taylor Series Best Linear and Quadratic Approximations; estimates on the errors of such approximations.

## Problem and Solution Sheets

### Question Sheets

In the nine question sheets there are a total of 95 questions with 54 additional questions. These questions reinforce the material in the lectures.

### Solution Sheets

 Solution Sheet 1 Solution Sheet 1 Additional questions Solution Sheet 2 Solution Sheet 2Additional Questions Solution Sheet 3 Solution Sheet 3Additional Questions Solution Sheet 4 Solution Sheet 4Additional Questions Solution Sheet 5 Solution Sheet 5Additional Questions Solution Sheet 6 Solution Sheet 6Additional Questions Solution Sheet 7 Solution Sheet 7Additional Questions Solution Sheet 8 Solution Sheet 8Additional Questions Solution Sheet 9 Differential forms Solution Sheet 9 Additional questions

## Background Notes

### Analysis

A very good knowledge of the results and methods of real analysis, as found in MATH20101 or MATH20111, is required. The present course will take results from those courses, such as the Inverse Function Theorem, and generalise them to vector valued functions of severable variables.

### Linear Algebra

I expect you remember the ideas and results of Linear Algebra found in MATH10202 and MATH10212, such as linear maps being represented by matrices, vector spaces and their bases. I will also rely on results not found there, such as the Spectral Theorem for Symmetric matrices. I would advise you to research this result, though appropriate notes will be given.

### Differential Geometry

A number of years ago I gave a short course, 8 weeks, on Differential Geometry. You can find the notes here, you might find them interesting. The Calculus of Several Variables and Differential Geometry courses have different goals. In Differential Geometry we are not interested in the largest set of differentiable functions, instead they are all considered to have continuous derivatives of all orders, i.e. are smooth functions. Also in Differential Geometry all surfaces are manifolds, unions of patches. And notation is different, particularly for the directional derivative. The notation in Differential Geoemtry eases generalisations from tangent vectors to vector fields.

 Chapter 1 Chapter 2 Chapter 3 Chapter 4

The main reference for these notes is Elementary Differential Geometry, Barrett O'Neill, Academic Press, 1966.