Last updated 18^{th} April 2018
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:
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 UEQsThe coursework test will be on Thursday, 15th March at 11.00am in Simon Theatre B.
Please turn up in good time, leaving your bags at front of the rooms containing your silent phones.
You are not allowed, and do not need, calculators. It is a closed book exam so no notes.
There will be five questions, on definitions, statements of results and calculations as seen on my problem sheets. No proofs required - they will be left for the exam. The material goes up to the Chain Rule. The test will last 45 minutes.
Resist the temptation to look at the answers of students sitting close by.
9.00 | 10.00 | 11.00 | 12.00 | 1.00 | 2.00 | 3.00 | 4.00 | 5.00 | |
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Monday | Feedback Tutorial Zochonis B26 |
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Tuesday | Lecture Stopford Th 1 |
Feedback Tutorial Samuel Alexander A116 |
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Wednesday | |||||||||
Thursday | Feedback Tutorial Samuel Alexander A102 |
Lecture Stopford Th 1 |
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Friday |
Exam Papers | Solutions and Feedback |
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June 2015 | 2015 Solutions |
June 2016 | 2016 Solutions 2016 Feedback |
June 2017 | 2017 solutions & Feedback |
Notes | Contents |
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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? C^{1} 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 C^{1}; 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 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. |
3. Tangent Spaces | 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; 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. | |
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 |
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Higher Order Derivatives: Higher order partial derivatives; The Hessian Matrix. | |
Taylor Series Best Linear and Quadratic Approximations; estimates on the errors of such approximations. |
In the nine question sheets there are a total of 95 questions with 54 additional questions. These questions reinforce the material in the lectures.
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.
Reading List for MATH20132.