The official source of material for this course is the syllabus page, but here I will give more details of the course and its examination procedures.

UEQ: Feedback on the Exam

Here are the questions, model solutions and feedback on your answers.

Feedback on Exam

UEQ: Feedback on feedback

I have made a few comments on the feedback you have given me and the course. One person wrote "Really, I never knew proofs get this long." I'm sorry, there is little I can do about this, it is the nature of the beast. If you stay on into the fourth year why not take my course on Analytic Number Theory, where we take half the course to prove the Prime Number Theorem.

Feedback on UEQs

Past & Present exam papers

Exam Papers Solutions and
June 2014 2014 Solutions
June 2015 2015 Solutions
June 2016 2016 Solutions 2016 Feedback

Lecture Notes

The lecture notes will be made available shortly before the material is presented in lectures.

Notes Contents
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.
Appendix Uniqueness of Limit; Example of Limit; Sandwich Rule; Composition Results.
Differentiation of functions of several variables: Directional derivatives; Partial Derivatives; Differentiable implies Continuous I; Vector-valued Linear Functions; Scalar-valued Linear Functions; Frechet Derivative; Best Affine Approximation; Differentiable implies Continuous II; Derivative exists implies directional derivative exists; Jacobian Matrix; Special case: Gradient vector; Jacobian Matrix revisited; Use of the Jacobian matrix; When is a function differentiable?; Examples; Product and Quotient functions; Rules for differentiation; The Chain Rule; The Chain Rule: Special cases; Products and Quotients of Scalar-valued functions; the Inverse function.
Appendix Partial and directional derivatives; Directional derivative implies directional continuity; Bounds on Linear maps; Derivative is unique; the Best Affine Approximation is unique; Mean Value Theorem; Differentiable does not implies C1; Earlier example revisited; Rules of Differentiation.
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; Level sets are graphs (locally): Implicit Function Theorem; Parametric sets are graphs (locally): Inverse Function Theorem; Manifolds; Tangent Spaces and Planes; Tangent Space for a graph; Tangent Space for an Parametric set; Tangent Space for a Level Set;
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 R^3; 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.
Appendix on Tangent Spaces Tangent Spaces for a surface in R^3 with illustrations.
Proof of the Implicit Function Theorem: by Induction.
Extremal Values: Extremal Values and Lagrange multipliers;
Appendix Examples including the GM-AM inequality & the Cauchy-Schwarz inequality along with their extensions.
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; These later topics are not covered in 2017 and not examinable: Stoke's Theorem (statement of); Green's Theorem (statement of); Differentiable k-forms; products and derivatives of k-forms; integration of a k-form over a manifold.
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.


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Problem and Solution Sheets

Question Sheets

Solution Sheets

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Background Notes


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.

Recommended Texts

Reading List for MATH20132.