# VECTOR CALCULUS - 2024/5

Module code: MAT1005

## Module Overview

The main focus of this module is on multivariable calculus in 2 and 3 dimensions, and vector calculus. This extends knowledge developed in A-levels and in the Year 1 module Calculus on the differentiation and integration of functions of a single variable, and provides the necessary ground work for Years 2 and 3 modules, such as Curves and Surfaces, Linear PDEs, Fluid Mechanics and Functions of a Complex Variable.

### Module provider

Mathematics & Physics

PRINSLOO Andrea (Maths & Phys)

### Module cap (Maximum number of students): N/A

Independent Learning Hours: 52

Lecture Hours: 44

Seminar Hours: 5

Guided Learning: 5

Captured Content: 44

Semester 2

MAT1030 Calculus

## Module content

Indicative content includes:

Vectors, and the Geometry of Curves and Surfaces in 2 and 3 dimensions:

• A review of the algebra and geometry of vectors, scalar and vector products, triple products, and rotations of vectors in 2 and 3 dimensions;

• Vector and cartesian equations of straight lines and planes in 2 and 3 dimensions;

• Equations of conic sections and quadric surfaces.

Multivariable Differential Calculus in 2 and 3 dimensions:

• Continuity of functions of 2 and 3 variables;

• Partial derivatives, and the differentiability of functions of 2 and 3 variables; the tangent plane at a point to a surface with height a function of 2 variables;

• Taylor expansions of functions of 2 and 3 variables, and the classification of stationary points;

• Chain rule for functions of 2 and 3 variables;

• Method of Lagrange multipliers for functions of 2 and 3 variables.

Double and Triple Integrals:

• Double and triple integrals in cartesian coordinates;

• Double integrals under coordinate transformations, including to polar coordinates;

• Triple integrals under coordinate transformations, including cylindrical coordinates and spherical polar coordinates.

Vector Calculus:

• Scalar and vector fields in 2 and 3 dimensions;

• Differential operators, including div, grad and curl, and their properties;

• Line and surface integrals;

• Green’s theorem, Stoke’s theorem and the Divergence Theorem.

## Assessment pattern

Assessment type Unit of assessment Weighting
School-timetabled exam/test In-semester test (50 minutes) 25
Examination Exam (2 hours) 75

N/A

## Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate understanding through the solution of unseen and similar to seen problems in the test and exam, which relate to:

• Geometric problems with vectors in 2 and 3 dimensions.

• Problems in multivariable differential calculus in 2 and 3 dimensions.

• Double and triple integrals.

• Scalar and vector fields, and differential operators.

• Line and surface integrals.

• Green’s theorem, Stoke’s theorem and the Divergence Theorem.

Thus, the summative assessment for this module consists of:

• One two hour examination at the end of the semester – worth 75% of the module mark.

• One in-semester test – worth 25% module mark.

Formative assessment and feedback

Students receive written feedback via a number of marked coursework assignments over an 11 week period. Additional verbal feedback will be provided by the lecturer or a tutor at seminars.

## Module aims

• Revisit the geometry of vectors, and curves and surfaces in 2 and 3 dimensions.
• Introduce students to multivariable calculus in 2 and 3 dimensions, including:
1) Differential calculus of functions of 2 and 3 variables;
2) Double and triple integrals.
• Introduce students to vector calculus in 3 dimensions, including:
1) Differential operators, such as div, grad and curl;
2) Line and surface integrals;
3) Green’s theorem, Stoke’s theorem and the Divergence Theorem.
• Enable students to solve problems related to each of these topics.

## Learning outcomes

 Attributes Developed 001 Demonstrate a working knowledge of vectors, and their uses in solving geometric problems in 2 and 3 dimensions. KCT 002 In the area of multivariable differential calculus, students will be able to do some or all of the following: 1) Understand what is meant by the continuity of functions of 2 and 3 variables. 2) Calculate partial derivatives of functions of 2 and 3-variables. 3) Find the tangent plane at a point to a surface, with height given by a function of 2 variables; and understand what is meant by the differentiability of functions of 2 and 3 variables. 4) Apply the chain rule for functions of 2 and 3 variables. 5) Classify the stationary points of functions of 2 and 3 variables; and solve both unconstrained and constrained optimization problems. KCT 003 In the area of multiple integration, students will be able to do some or all of the following: 1) Compute double and triple integrals. 2) Perform coordinate transformations. KCT 004 In the area of vector calculus, students will be able to do some or all of the following: 1) Calculate the action of differential operators on scalar and vector fields. 2) Compute line and surface integrals in 3 dimensions. 3) Apply Greens theorem, Stokes theorem and the Divergence Theorem. KCT

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

## Methods of Teaching / Learning

The learning and teaching strategy is designed to:

• Extend students’ knowledge of vectors, and geometry in 2 and 3 dimensions.

• Provide students with a detailed introduction to multivariable calculus in 2 and 3 dimensions, and vector calculus in 3 dimensions.

• Experience (through demonstration) of the methods used to interpret, understand and solve problems in multivariable calculus and vector calculus.

The learning and teaching methods include:

• 4 x 1 hour lectures per week x 11 weeks – material and examples supplementing the module notes will be presented on blackboards/whiteboards, with Q & A opportunities for students.

• 1 x 1 hour seminar in alternate weeks x 5 weeks – a guided discussion of solutions to problem sheets, provided to and worked on by students in advance.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

Upon accessing the reading list, please search for the module using the module code: MAT1005

## Programmes this module appears in

Programme Semester Classification Qualifying conditions
Economics and Mathematics BSc (Hons) 2 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics with Statistics MMath 2 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics with Statistics BSc (Hons) 2 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics BSc (Hons) 2 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics with Music BSc (Hons) 2 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics MMath 2 Compulsory A weighted aggregate mark of 40% is required to pass the module
Financial Mathematics BSc (Hons) 2 Compulsory A weighted aggregate mark of 40% is required to pass the module

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2024/5 academic year.