## MAS111 Mathematics Core II

 Semester 2, 2017/18 20 Credits Lecturer: Dr Frazer Jarvis Home page (also MOLE) Timetable Reading List 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;
• Functions of two variables and partial differentiation; the Taylor series; tangents and normals; Jacobians.
• Multivariable calculus: further partial differentiation and the Chain rule; integration as area under a graph; two-dimensional integration; double integrals.

## 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

44 lectures, 11 tutorials

## Assessment

One formal 2 hour exam. All questions compulsory; format varies. (55 marks)

Five quizzes in the Problems Classes/Practicals. (5 marks)

## 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 R2R2, 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 RnRm, 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:R2R, 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.
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:R2R in terms of eigenvalues of the Hessian.
9. Series
Convergence of series. Radius of convergence. Integration as a limit of summations.
10. 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.
11. 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/(n2)].

Type Author(s) Title Library Blackwells Amazon
C Ross L. Finney, George B. Thomas, Jr. Calculus 517 (F) Blackwells Amazon
C M. Anthony and M. Harvey Linear Algebra: Concepts and Methods Blackwells Amazon
C R.B.J.T. Allenby Linear Algebra Blackwells Amazon
C Robert Smith, Roland Minton Calculus 515 (S) 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

 Mon 12:00 - 12:50 tutorial (group 2) (even weeks) 38 Mappin Street Room 102 Mon 12:00 - 12:50 tutorial (group 20) (even weeks) Hicks H7 Mon 12:00 - 12:50 tutorial (group 21) (even weeks) Hicks Seminar Room F30 Mon 12:00 - 12:50 tutorial (group 28) (even weeks) Hicks Seminar Room F35 Mon 12:00 - 12:50 tutorial (group 3) (even weeks) 38 Mappin Street Room 103 Mon 12:00 - 12:50 tutorial (group 31) (even weeks) Hicks I12 Mon 12:00 - 12:50 tutorial (group 34) (even weeks) K14 Hicks Building Mon 12:00 - 12:50 tutorial (group 4) (even weeks) 38 Mappin Street Room 103 Mon 12:00 - 12:50 tutorial (group 5) (even weeks) 38 Mappin Street Room 202 Mon 12:00 - 12:50 tutorial (group 6) (even weeks) Hicks I14 Mon 12:00 - 12:50 tutorial (group 7) (even weeks) 38 Mappin Street Room 204 Mon 14:00 - 14:50 tutorial (group 10) (even weeks) Hicks J6c Mon 14:00 - 14:50 tutorial (group 11) (even weeks) 38 Mappin Street Room 203 Mon 14:00 - 14:50 tutorial (group 12) (even weeks) 38 Mappin Street Room 204 Mon 14:00 - 14:50 tutorial (group 14) (even weeks) 38 Mappin Street Room G01 Mon 14:00 - 14:50 tutorial (group 8) (even weeks) Hicks J15 Tue 11:00 - 11:50 tutorial (group 16) (even weeks) 38 Mappin Street Room 202 Tue 11:00 - 11:50 tutorial (group 17) (even weeks) 38 Mappin Street Room 203 Tue 11:00 - 11:50 tutorial (group 18) (even weeks) 38 Mappin Street Room 204 Tue 11:00 - 11:50 tutorial (group 19) (even weeks) 38 Mappin Street Room G01 Wed 13:00 - 13:50 lecture Students Union Auditorium Thu 10:00 - 10:50 tutorial (group 22) (even weeks) Hicks I7 Thu 10:00 - 10:50 tutorial (group 23) (even weeks) 38 Mappin Street Room 103 Thu 10:00 - 10:50 tutorial (group 24) (even weeks) 38 Mappin Street Room 202 Thu 10:00 - 10:50 tutorial (group 26) (even weeks) 38 Mappin Street Room 204 Thu 10:00 - 10:50 tutorial (group 27) (even weeks) 38 Mappin Street Room 203 Thu 10:00 - 10:50 tutorial (group 35) (even weeks) Hicks Seminar Room F30 Thu 10:00 - 10:50 tutorial (group 36) (even weeks) Hicks Seminar Room F24 Thu 10:00 - 10:50 lab session (group A) (odd weeks) Hicks Seminar Room F20 Thu 10:00 - 10:50 lab session (group B) (odd weeks) Hicks Seminar Room F24 Thu 10:00 - 10:50 lab session (group C) (odd weeks) Hicks Seminar Room F38 Thu 11:00 - 11:50 tutorial (group 29) (even weeks) 38 Mappin Street Room 202 Thu 11:00 - 11:50 tutorial (group 30) (even weeks) 38 Mappin Street Room 203 Thu 11:00 - 11:50 tutorial (group 32) (even weeks) 38 Mappin Street Room 103 Thu 11:00 - 11:50 tutorial (group 33) (even weeks) 38 Mappin Street, Tutorial Room 9 Thu 11:00 - 11:50 tutorial (group 38) (even weeks) Hicks Lecture Theatre 4 Thu 11:00 - 11:50 lab session (group D) (odd weeks) Hicks Seminar Room F20 Thu 11:00 - 11:50 lab session (group E) (odd weeks) Hicks Seminar Room F24 Thu 11:00 - 11:50 lab session (group F) (odd weeks) Hicks Seminar Room F38 Thu 12:00 - 12:50 lecture Richard Roberts Auditorium Thu 15:00 - 15:50 tutorial (group 15) (even weeks) Hicks J26 Thu 15:00 - 15:50 tutorial (group 25) (even weeks) 38 Mappin Street Room 204 Thu 15:00 - 15:50 tutorial (group 37) (even weeks) 38 Mappin Street Room 203 Thu 15:00 - 15:50 tutorial (group 39) (even weeks) 38 Mappin Street Room G01 Thu 15:00 - 15:50 tutorial (group 40) (even weeks) 38 Mappin Street Room 106 Thu 15:00 - 15:50 tutorial (group 41) (even weeks) 38 Mappin Street Room 102 Thu 15:00 - 15:50 tutorial (group 42) (even weeks) 38 Mappin Street Room 202 Thu 15:00 - 15:50 tutorial (group 43) (even weeks) Hicks Seminar Room F24 Thu 15:00 - 15:50 lab session (group G) (odd weeks) Hicks Seminar Room F24 Thu 15:00 - 15:50 lab session (group H) (odd weeks) Hicks Seminar Room F41 Fri 09:00 - 09:50 lecture Richard Roberts Auditorium Fri 11:00 - 11:50 lecture Richard Roberts Auditorium