## MAS211 Advanced Calculus and Linear Algebra

 Semester 1, 2021/22 20 Credits Lecturer: Prof Neil Dummigan Timetable Reading List Aims Outcomes Teaching Methods 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 MAS346 Groups and Symmetry MAS348 Game Theory MAS350 Measure and Probability MAS420 Signal Processing MAS423 Advanced Operations Research

## Outline syllabus

1. Introduction to triple integrals.
2. Determinants.
3. Coordinate changes.
4. Subspaces of Rn
5. Linear maps from Rn to Rm.
6. Derivatives of general maps from Rn to Rm.
7. Line integrals in the plane and space.
8. Gradient of a scalar function, and stationary points of functions.
9. Quadratic forms, max and min in several variables.
10. Divergence and curl of vector fields.
11. Surface integrals.
12. Introduction to Fourier series.

## Aims

To introduce various techniques of Advanced Calculus and Linear Algebra

## Learning outcomes

1. Calculate determinants.
2. Use the rank/nullity theorem in the calculation of dimensions.
3. Find the rank of matrices.
4. Find the inverse of a square matrix (if it exists).
5. Understand and use curvilinear coordinate systems in the plane and space.
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.

## Teaching methods

44 lectures, 11 tutorials, 0 practicals.

44 lectures, 11 tutorials

## Assessment

Formal written examination.

## Full syllabus

1. Introduction to triple integrals. (2 lectures)
Easy examples. Average values.

2. Determinants. (4 lectures)
Determinants for square matrices of any size. Computation via Laplace expansion or row reduction. Multiplicative property. Criterion for invertibility. Relation to scaling and orientation in two and three dimensions.
3. Coordinate changes. (4 lectures)
Linear coordinate changes, polar coordinates, spherical and cylindrical polar coordinates. Local approximate linearity for increments. Local scaling of volume. Jacobian matrix and determinant. Application to triple integrals. General coordinate changes.
4. Subspaces of Rn. (3 lectures)
Explicit and implicit descriptions of subspaces of Rn (spans and null spaces). Existence of a basis for any subspace. Dimension of a subspace.
5. Linear maps from Rn to Rm. (4 lectures)
Image and kernel. Rank-nullity theorem. Affine maps.
6. Derivatives of general maps from Rn to Rm. (5 lectures)
Examples: scalar fields, parametrised curves and surfaces. Zero sets, level sets, images, graphs. Derivative matrices, examples including gradient, velocity, Jacobian matrices and complex derivatives. Local linearity for increments, tangent space to a graph. General Chain Rule.

7. 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.
8. Gradient of a scalar function and stationary points of functions (5 lectures)
Gradient as a vector corresponding to derivative, directional derivative. Local extrema and saddle points, classifying stationary points; constrained maxima and minima; Lagrangian multipliers.
9. Quadratic forms, max and min in several variables (5 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.
10. Divergence and Curl of vector fields (2 lectures)
Definition of the curl and the divergence of a vector field; the del and Laplace operators.
11. Surface integrals (3 lectures)
Smooth surfaces; surface integrals; Gauss's theorem; Stokes' theorem.
12. Introduction to Fourier Series (3 lectures)