MATH20132.html

Course Material

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.

Lecture Notes

Notes	Contents
Continuity	Functions of Several Variables: Notation; Limits of vector-valued functions; Scalar-valued examples; Sandwich Rule; Directional limits; Limits along curves; Continuity of vector-valued functions; Continuity Laws; Composition Laws. Appendix Uniqueness of Limit; Example of Limit; Directional limits; Composition Results.
Differentiation	Differentiation of functions of several variables: Directional derivatives; Partial Derivatives; Differentiable implies Continuous; Vector-valued Linear Functions; Fréchet Derivative. Fréchet Differentiable implies Continuous; Derivative exists implies directional derivative exists. Jacobian Matrix; Special cases: functions of one variable and scalar valued functions. When is a function differentiable? C¹ functions are Fréchet differentiable. Examples; Product and Quotient functions; The Chain Rule; Rules for differentiation; Special cases; the Inverse function. 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.
Surfaces A Surfaces B	Surfaces and the Implicit Function Theorem: A surface as a graph; a surface as an Image set, i.e. given parametrically; a surface as a Level set, i.e. given Implicitly; a graph is a level set; converses. Linear Algebra, Vector Subspaces & planes. Level sets are graphs (locally): Implicit Function Theorem; Parametric sets are graphs (locally): Inverse Function Theorem; Manifolds. Tangent Spaces and Tangent Planes; Tangent Space for a graph, for a Level Set and for a Parametric 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; ℝⁿ or ℝ^r x ℝ^n-r?; Surface of a unit ball in ℝ³; Permuting the coordinate functions in a parametric set; Permuting the variables in a level set; Background Linear Algebra; Dimension; Surfaces as level sets f^-1(0). Which f ?; Proof of the Inverse Function Theorem. Appendix on Tangent Spaces Tangent Spaces for a surface in ℝ³ with illustrations.
Implicit Function Theorem	Proof of the Implicit Function Theorem: by Induction.
Extremal Values	Extremal Values: Extremal Values and Lagrange multipliers; Appendix Examples including the GM-AM inequality & the Cauchy-Schwarz inequality along with their extensions.
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). 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.
Forms & Integration	Higher Derivatives: The Hessian Matrix; Taylor's Theorem; Surfaces; examples of local minima, maxima and saddle points; Tests for definiteness or otherwise. Appendix Further worked example; Tests for definiteness or otherwise of matrices.

Notes

Contents

Continuity

Functions of Several Variables: Notation; Limits of vector-valued functions; Scalar-valued examples; Sandwich Rule; Directional limits; Limits along curves; Continuity of vector-valued functions; Continuity Laws; Composition Laws.

Appendix Uniqueness of Limit; Example of Limit; Directional limits; Composition Results.

Differentiation

Differentiation of functions of several variables: Directional derivatives; Partial Derivatives; Differentiable implies Continuous; Vector-valued Linear Functions; Fréchet Derivative.
Fréchet Differentiable implies Continuous; Derivative exists implies directional derivative exists.
Jacobian Matrix; Special cases: functions of one variable and scalar valued functions.
When is a function differentiable? C¹ functions are Fréchet differentiable.
Examples; Product and Quotient functions; The Chain Rule; Rules for differentiation; Special cases; the Inverse function.

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.

Surfaces A
Surfaces B

Surfaces and the Implicit Function Theorem: A surface as a graph; a surface as an Image set, i.e. given parametrically; a surface as a Level set, i.e. given Implicitly; a graph is a level set; converses.
Linear Algebra, Vector Subspaces & planes.
Level sets are graphs (locally): Implicit Function Theorem; Parametric sets are graphs (locally): Inverse Function Theorem;
Manifolds.
Tangent Spaces and Tangent Planes; Tangent Space for a graph, for a Level Set and for a Parametric 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; ℝⁿ or ℝ^r x ℝ^n-r?; Surface of a unit ball in ℝ³; Permuting the coordinate functions in a parametric set; Permuting the variables in a level set; Background Linear Algebra; Dimension; Surfaces as level sets f^-1(0). Which f ?; Proof of the Inverse Function Theorem.

Appendix on Tangent Spaces Tangent Spaces for a surface in ℝ³ with illustrations.

Implicit Function Theorem

Proof of the Implicit Function Theorem: by Induction.

Extremal Values

Extremal Values: Extremal Values and Lagrange multipliers;

Appendix Examples including the GM-AM inequality & the Cauchy-Schwarz inequality along with their extensions.

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). 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.

Forms & Integration

Higher Derivatives: The Hessian Matrix; Taylor's Theorem; Surfaces; examples of local minima, maxima and saddle points; Tests for definiteness or otherwise.

Appendix Further worked example; Tests for definiteness or otherwise of matrices.

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.

Problem Sheet 1 Limits	Problem Sheet 2 Continuity
Problem Sheet 3 The Directional Derivative	Problem Sheet 4 The Frechet Derivative I
Problem Sheet 5 C¹-functions and more	Problem Sheet 6 Surfaces as level sets, as image sets, Graphs, Implicit function Theorem
Problem Sheet 7 Inverses, Best Affine Approximations, Tangent Spaces for graphs	Problem Sheet 8 Tangent Planes to level sets & Image sets
Problem Sheet 9 Lagrange’s Method	Question Sheet 10 Differential Forms
Problem Sheet 11 Higher order forms

Solution Sheet 1	Solution Sheet 2	Solution Sheet 3	Solution Sheet 4
Solution Sheet 5	Solution Sheet 6	Solution Sheet 7	Solution Sheet 8
Solution Sheet 9	Solution Sheet 10	Solution Sheet 11

MATH20132

Calculus of Several Variables

CONTACT: Dr. M D Coleman. Room 1.109 Alan Turing Building

Course Material

Lecture Notes

Problem and Solution Sheets

Problem Sheets

Solution Sheets

Background Notes

Analysis

Linear Algebra

Differential Geometry

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