## MAS211 Advanced Calculus and Linear Algebra

Note: This is an old module occurrence.

You may wish to visit the module list for information on current teaching.

 Semester 1, 2015/16 20 Credits Lecturer: Dr Kirill Mackenzie Home page Timetable Reading List Aims Outcomes Assessment Full Syllabus

`Advanced Calculus and Linear Algebra' are basic to most further work in pure and applied mathematics and to much of statistics. This course provides the basic tools and techniques and includes sufficient theory to enable the methods to be used in situations not covered in the course. The material in this course is essential for further study in mathematics and statistics.

Prerequisites: MAS111 (Mathematics Core II)

The following modules have this module as a prerequisite:

 MAS212 Scientific Computing and Simulation MAS220 Algebra MAS221 Analysis MAS222 Differential Equations MAS223 Statistical Inference and Modelling MAS275 Probability Modelling MAS279 Career Development Skills MAS280 Mechanics and Fluids MAS301 Group Project MAS322 Operations Research MAS325 Mathematical Methods MAS330 Topics in Number Theory MAS332 Complex Analysis MAS334 Combinatorics MAS336 Differential Geometry MAS341 Graph Theory MAS342 Applicable Analysis MAS345 Codes and Cryptography MAS346 Groups and Symmetry MAS348 Game Theory MAS350 Measure and Probability MAS420 Signal Processing MAS423 Advanced Operations Research MAS441 Optics and Symplectic Geometry MAS445 Mathematics (Numerical Methods)

## Outline syllabus

Curvilinear coordinates. Continuous differentiability and C1. The derivative as a matrix. Chain rules in matrix form.

Linear maps: vector subspaces of Rp, rank and nullity, orthogonal complements, adjoints and transposes.

Review of double integrals. Triple integrals. Change of variables in double and triple integrals. Jacobians. Applications.

Determinants of arbitrary order: conditions which determine the determinant. Methods of calculation. Invertibility of matrices and methods for finding inverse. Elementary matrices. Diagonalization; powers of matrices.

Line integrals in the plane and space. Calculation using parametrised curves. Green's theorem in the plane. Integrals of exact differentials in the plane. Potential functions. Criterion for exactness. Independence of path.

Gradient as vector corresponding to derivative. Surface integrals over projectable patches. Notion of vector field. Divergence and potentials, Divergence theorem. Statement of Stokes' Theorem.

Orthogonal and symmetric matrices. Diagonalization of quadratic forms. Classification of critical points. Constrained maxima and minima, Lagrange multipliers.Chain rules for second derivatives of smooth functions if time permits.

## Office hours

Friday 2pm to 2.50 or by appointment

## Aims

To introduce various techniques of Advanced Calculus and Linear Algebra

## Learning outcomes

1. Calculate determinants of constants and of functions including n×n examples for n = 4, 5 and general n.
2. Use the rank/nullity theorem in the calculation of dimensions.
3. Find the rank of matrices of constants and of functions.
4. Find the inverse of a square matrix (if it exists).
5. Understand and use curvilinear coordinate systems.
6. Understand and use the Chain Rules for maps of several variables.
7. Find the derivative matrix of a vector-valued function.
8. Compute double and triple integrals and evaluate them using substitutions.
9. Compute line and surface integrals and use Green's theorem to evaluate line integrals and areas.
10. Calculate line integrals along parametrised curves.
11. Evaluate line integrals of exact differentials by changing the path or finding a potential function.
12. Apply Stokes' and the Divergence Theorem to evaluate multiple integrals.
13. Find and classify the critical points of functions of two (or more) variables.
14. Find the canonical form of a quadratic form.
15. Use Lagrange multipliers to find critical points subject to constraints.

44 lectures, 10 tutorials

## Assessment

Formal 2.5 hour written examination with all questions to be attempted

## Full syllabus

1. Curvilinear coordinates. (6 lectures)
Curvilinear coordinates as maps: parabolic, spherical, cylindrical, and other standard examples. Derivatives of maps RpRq as matrices of partials. Use of the MAPLE command plot3d. Continuous differentiability and C1. Chain rules in matrix form, including general case.

2. Linear maps. (6 lectures)
Vector subspaces of Rp. Linear maps and matrices. Kernel and image. Rank and nullity. Orthogonal complements. Adjoints and transposes.
3. Double and triple integrals (4 lectures)
Review of double integrals. Determinant as ratio of areas/volumes.Triple integrals over simple regions. Change of variables in double and triple integrals. Jacobians. Applications.
4. Determinants (6 lectures)
Determinants of p×p matrices. Elementary matrices. Invertibility and inverses of square matrices. Methods of calculation.
5. Line integrals in the plane and space (4 lectures)
Line integrals and work done. Basic properties. Calculating line integrals using parametrised curves. Green's theorem in the plane, including application to calculating areas. Integrals of exact differentials in the plane, finding potential functions. Criterion for exactness. Independence of path.
6. Subspaces and normal forms (4 lectures)
Subspaces of Rn. Spans. Canonical bases. Sums and intersections of subspaces. Diagonalizability. Powers of matrices. Introduction to normal forms for nondiagonalizable matrices.
7. Gradient, divergence, curl (6 lectures)
Gradient as vector corresponding to derivative. Surface integrals over projectable patches. Notion of vector field. Divergence and potentials, Divergence theorem. Statement of Stokes' Theorem.
8. Quadratic forms, max and min in several variables (4 lectures)
Orthogonal and symmetric matrices. Diagonalization of quadratic forms. Classification of critical points with description of n variable case, and details for n = 2,3. Constrained maxima and minima, Lagrange multipliers. Chain rules for second derivatives of smooth functions if time permits.

## Reading list

Type Author(s) Title Library Blackwells Amazon
B D.W. Jordan and P. Smith Mathematical Techniques 10 (J) Blackwells Amazon
B E. Kreyszig Advanced Engineering Mathematics Q 510 (K) Blackwells Amazon
B M.R. Spiegel Advanced Calculus Q 515.076 (W) Blackwells Amazon

(A = essential, B = recommended, C = background.)

Most books on reading lists should also be available from the Blackwells shop at Jessop West.

## Timetable

 Tue 09:00 - 09:50 tutorial (group 407) Hicks Seminar Room F28 Wed 09:00 - 09:50 tutorial (group 402) Hicks Lecture Theatre D Fri 12:00 - 12:50 lecture Arts Tower Lecture Theatre 4