## 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, 2017/18 20 Credits Lecturer: Dr Kirill Mackenzie Home page 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 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

## Outline syllabus

Curvilinear coordinates in the plane. Smooth functions and maps. The derivative as a matrix. Jacobians. Chain rules in matrix form. Inverse maps. Review of double integrals. Change of variables in double integrals.

Curvilinear coordinates in three dimensions. Spherical and cylindrical polars. Oblate spheroidal coordinates.

Linear maps and vector subspaces of Rp. Linear independence and spanning sets. Bases and dimension. Linear maps. Rank and nullity.

Determinants and inverse matrices. Cross products. Volumes as determinants and triple products. Finding matrix inverses.

Triple integrals. Change of variables in triple integrals.

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

Tuesday 2pm (Nick Monk), Fridays 2pm (Kirill Mackenzie)

## Aims

To introduce various techniques of Advanced Calculus and Linear Algebra

## Learning outcomes

1. Calculate determinants of constants and of functions.
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 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, 10 tutorials, 0 practicals.

44 lectures, 10 tutorials

## Assessment

Formal 2.5 hour written examination with all questions to be attempted

## Full syllabus

1. Curvilinear coordinates and smooth maps. (4 lectures approx.)
Curvilinear coordinates in the plane. Smooth functions and maps. The derivative of maps from R2 to R2 as matrices of partials. Jacobians. Chain rules in matrix form. Inverse maps. Review of double integrals. Change of variables in double integrals.

2. Curvilinear coordinates in three dimensions. (5 lectures approx.)
Curvilinear coordinates in R3. Spherical polars. Cylindrical polars. Oblate spheroidal coordinates. The derivative of maps from R3 to R3 as matrices of partials. Jacobians. Chain rules in matrix form. Inverse maps. General Chain Rule.
3. Linear maps and vector subspaces. (5 lectures approx.)
Linear maps from Rp to Rq and their matrices. Vector subspaces of Rp. Linear independence and spanning sets. Bases and dimension. Nullspace (kernel) and image. Rank and nullity.
4. Determinants and inverse matrices. (4 lectures approx.)
Cross products. Volumes as determinants and as triple products. Finding matrix inverses.
5. Triple integrals (4 lectures approx.)
Triple integrals over normal regions. Change of variables in triple integrals. Applications.
6. 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.
7. 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.
8. 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.
9. 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.