MA14510 Mathematical Methods and Modelling 2
Introductory description
This module is centred on intuitive geometric and physical concepts such as length, area, volume, curvature, mass, circulation and flux and their translation into mathematical formulas. The focus is on the practical calculation of these formulas and their application to various problems.
Module aims
To introduce and apply methods and techniques from vector calculus.
Outline syllabus
This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.
 Basic geometry of R^n with focus on R^3: vectors, dotproducts, angles, components in given direction (in an orthonormal basis), equations of planes.
 Differential geometry of curves: parametrisation, tangents, integration, arclength, GradshteynRyzhik. [No Frenet frames or crossed products]
 Differential calculus of scalar functions of several variables: partial derivative, chain rule, change of coordinates, gradient, directional derivative, tangent planes, level sets. [No higher order derivatives or critical points]
 Integral calculus of scalar functions of several variables: multiple integration, rectangles, boxes, 2D integrals as Riemann sums, determinants, change of variables, Jacobian, integration in other coordinates (polar, cylindrical, spherical).
 Differential calculus of vector functions of several variables: crossed product, vector fields, divergence, curl, nabla, algebraic identities, Laplace, expression in other coordinates (polar, cylindrical, spherical), parametrisation of surfaces, tangent planes, normal.
 Integral calculus of vector functions of several variables: surface area, surface integral, flux, divergence theorem, line integrals, work and potential energy, circulation, Green’s Theorem, Stokes’ theorem.
Learning outcomes
By the end of the module, students should be able to:
 working knowledge of scalarvalued and vectorvalued functions of one or more variables
 understand parametric representations of curves and surfaces
 perform coordinate transformations
 demonstrate understanding of integral theorems relating line, surface and volume integrals and evaluate such integrals
 understand concepts and techniques related to the differentiation for functions from R^n to R^m, such as partial derivatives, gradient, Jacobian, directional derivative, divergence and curl
 get familiar with the Riemann integral of a multivariable function and its geometric interpretation
 be able to integrate functions over simple domains and justify the change of area and volume elements when converting coordinates
Indicative reading list
Stewart, J., Multivariable Calculus, Cengage Learning, 2011.
Marsden, J., Tromba, A. J. and Weinstein, A., Basic Multivariable Calculus, Springer 1993.
Strang, G., Calculus, WellesleyCambridge Press, 1991.
G.B. Thomas et al., Calculus and Analytic Geometry, AddisonWesley, 1969.
F.J. Flannigan and J.L. Kazdan, Calculus Two, SpringerVerlag, 1990
View reading list on Talis Aspire
Subject specific skills
This module will introduce students to connections between geometry, calculus and physical modelling, providing them with the skills to convert between reasoning physically and geometrically. Further, the understanding of and ability to manipulate concepts from multivariable calculus first developed here is crucial to understand many of the mathematical models used to describe the world around us.
Transferable skills
Along with developing problem solving skills and logical reasoning, a key aim of this module is building confidence in visualising and sketching geometric objects, enabling the development of spatial awareness and visual communication skills.
Study time
Type  Required 

Lectures  20 sessions of 1 hour (48%) 
Online learning (independent)  9 sessions of 1 hour (21%) 
Private study  13 hours (31%) 
Total  42 hours 
Private study description
Working on assignments, going over lecture notes, text books, exam revision.
Costs
No further costs have been identified for this module.
You do not need to pass all assessment components to pass the module.
Assessment group D
Weighting  Study time  

Assignments  15%  20 hours 
Inperson Examination  85%  38 hours 

Assessment group R
Weighting  Study time  

Inperson Examination  Resit  100%  

Feedback on assessment
Marked homework (both assessed and formative) is returned and discussed in smaller classes. Exam feedback is given.
Courses
This module is Core for:

UMAAGV17 Undergraduate Mathematics and Philosophy
 Year 1 of GV17 Mathematics and Philosophy
 Year 1 of GV17 Mathematics and Philosophy
 Year 1 of GV17 Mathematics and Philosophy

UPXAGF13 Undergraduate Mathematics and Physics (BSc)
 Year 1 of GF13 Mathematics and Physics
 Year 1 of GF13 Mathematics and Physics

UPXAFG31 Undergraduate Mathematics and Physics (MMathPhys)
 Year 1 of FG31 Mathematics and Physics (MMathPhys)
 Year 1 of FG31 Mathematics and Physics (MMathPhys)
This module is Optional for:
 Year 1 of USTAG300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics
 Year 1 of USTAG1G3 Undergraduate Mathematics and Statistics (BSc MMathStat)

USTAGG14 Undergraduate Mathematics and Statistics (BSc)
 Year 1 of GG14 Mathematics and Statistics
 Year 1 of GG14 Mathematics and Statistics

USTAY602 Undergraduate Mathematics,Operational Research,Statistics and Economics
 Year 1 of Y602 Mathematics,Operational Research,Stats,Economics
 Year 1 of Y602 Mathematics,Operational Research,Stats,Economics
This module is Option list B for:

UCSAG4G1 Undergraduate Discrete Mathematics
 Year 1 of G4G1 Discrete Mathematics
 Year 1 of G4G1 Discrete Mathematics
 Year 1 of UCSAG4G3 Undergraduate Discrete Mathematics