## MAS111 Mathematics Core II

Semester 2, 2021/22 | 20 Credits | ||||

Lecturer: | Dr Frazer Jarvis | Timetable | |||

Aims | Outcomes | Teaching Methods | Assessment | Full Syllabus |

This module aims to extend the material from MAS110. The central aims of this course will be to learn how to interpret the geometry of functions with more than one variable, solve systems of linear equations, and use calculus to understand the graphs of functions with several variables and the volumes that they bound. Material covered will include, but is not limited to, plane and solid geometry, matrix multiplication, linear equations, Gaussian elimination, graphs and level sets of functions with two variables, partial derivatives, volumes, and double integrals.

**Prerequisites:** MAS110 (Mathematics Core I)

The following modules have this module as a prerequisite:

MAS112 | Vectors and Mechanics |

MAS211 | Advanced Calculus and Linear Algebra |

MAS220 | Algebra |

MAS221 | Analysis |

MAS222 | Differential Equations |

## Outline syllabus

- Basic plane geometry in 2 and 3 dimensions; solution of simultaneous equations, both geometrically as intersections of planes, and algebraically via Gaussian reduction.
- Matrices: introduction; interpretation as linear maps and the multiplication rule; determinants; eigenvalues and eigenvectors; change of basis.
- Functions of two variables and partial differentiation; the Chain rule; the Taylor series; tangents and normals; Jacobians; higher partial derivatives; classification of stationary points.
- Multivariable integration: 1-dimensional integration as area under a graph; 2-dimensional integration as a volume; double integrals; change of variables.

## Aims

- To demonstrate techniques for representing lines and conics using co-ordinate geometry.
- To study systematic ways of solving simultaneous equations.
- To introduce matrices and matrix arithmetic.
- To develop students' skills in the solution of problems in matrix algebra and co-ordinate geometry.
- To introduce the ideas of determinants, eigenvalues and eigenvectors.
- To introduce series.
- To introduce the basic techniques of calculus of functions of more than one variable, and to gain expertise in calculating partial derivatives and double integrals, and in using the Chain Rule.

## Learning outcomes

- understand 2- and 3-dimensional geometry;
- solve linear equations;
- use matrices and understand them as linear maps;
- compute determinants and understand them as scaling factors;
- compute eigenvalues and eigenvectors;
- differentiate functions of more than one variable, including use of the Chain Rule;
- evaluate double integrals.

## Teaching methods

Lectures, Problem Solving/Example Classes; possibly videos (if required)

44 lectures, 11 tutorials

## Assessment

2021-22: One formal 2 hour exam. All questions compulsory; format varies (60 marks, worth 90%). Five Blackboard quizzes (10%).

## Full syllabus

**1. Geometry in two and three dimensions**

Coordinate systems in two and three dimensions. Lines in two dimensions, and basic properties. Spherical and cylindrical polar coordinates in three dimensions; planes and their intersections.

**2. Simultaneous equations**

Solving simultaneous equations in three variables. Gaussian and complete reduction. Row echelon form and reduced row echelon form. Linear independence and criteria for systems to have a unique solution.

**3. Matrices**

2×2 matrices as linear maps from

**R**

^{2}→

**R**

^{2}, multiplication of matrices as the composition of maps. Specific examples of matrices like rotations. 3x3 matrices. General matrix notation, addition and multiplication of matrices, matrices as linear maps

**R**

^{n}→

**R**

^{m}, inverse/identity matrices, isomorphisms. Elementary matrices/maps, solving systems of linear equations by Gaussian elimination, finding the inverse of a matrix using row operations.

**4. Determinants**

Determinants of 2x2 and 3x3 matrices. The determinant of a 2x2 matrix as an oriented area, and of a 3x3 matrix as an oriented volume. n×n determinants. The determinant of an n×n matrix, properties of the determinant like det(AB)=det(A)det(B), row operations etc.

**5. Eigenvalues and eigenvectors**

Eigenvalues and eigenvectors and geometric interpretation. Applications. Coupled differential equations.

**6. Functions of two variables and partial differentiation**

Functions f:

**R**

^{2}→

**R**, their graphs, level sets. Intersection of graphs with planes, partial derivatives, directional derivatives and graphical interpretation. Loci of planes, spheres, cones, ellipsoids, other simple objects. Normal vectors, tangent planes. Higher partial derivatives, equality of mixed derivatives, Taylor series. Small increments. The Chain Rule and its applications, including to Laplace's equation.

**7. Quadratic curves**

Conic sections as the intersection of cones and planes. Basic properties. Focus-directrix definitions and reflection properties. Hyperbolic functions, both as parametrising a hyperbola, and as interesting functions in their own right.

**8. Classification of stationary points of functions of two variables**

Quadratic forms of two variables, classification in terms of the discriminant. Characterisation of critical points for functions f:

**R**

^{2}→

**R**in terms of eigenvalues of the Hessian.

**9. Integration of functions of one variable**

Areas under graphs, integration of powers from first principles, average values. Fundamental Theorem of Calculus. Area of a circle, volume and surface area of a sphere. Arc length. Volumes and surface areas of revolution.

**10. Double integrals**

Review of the Fundamental Theorem of Calculus. Two-dimensional integrals as volumes under graphs, their evaluation by double integration, in either order. Integration by substitution. Change of variables, including to polar coordinates. The probabilistic integral and the sum ∑[1/(n

^{2})].

**11. Series (if time permits)**

Convergence of series. Radius of convergence. Integration as a limit of summations.