MATH2365 Vector Calculus

15 creditsClass Size: 400

Module manager: Dr Mike Evans
Email: R.M.L.Evans@leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2024/25

Pre-requisite qualifications

MATH1005 or MATH1010 or (MATH1050 and MATH1060) or (MATH1050 and MATH1331), or equivalent.

This module is approved as a discovery module

Module summary

Vector calculus is the extension of ordinary one-dimensional differential and integral calculus to higher dimensions. As such it provides the mathematical framework for the study of a wide variety of physical systems, such as fluid mechanics and electromagnetism that can be described by vector and scalar fields.

Objectives

On completion of this module, students should be able to:

a) calculate vector and scalar derivatives of vector and scalar fields using the grad, div and curl operators in Cartesian and in cylindrical and spherical polar coordinates;
b) use suffix notation to manipulate Cartesian vectors and their derivatives;
c) calculate multiple integrals in two and three dimensions including changing variables using Jacobians;
d) calculate line and surface integrals and use the various integral theorems.

Syllabus

1. Vector Calculus: grad, div, curl and the del operator. The directional derivative and Laplacian operators.
2. Suffix notation: representation of vectors and their products using suffix notation. The Kronecker delta and alternating tensors. Grad, div and curl in suffix notation. Use of suffix notation to manipulate products and combinations of vector derivatives.
3. Double and triple integrals. Change of order of integration for double integrals over non-rectangular domains. Transformation of coordinates: the Jacobian. Cylindrical and spherical polar coordinates.
4. Scalar line and surface integrals in 3 dimensional space. Parameterisation of lines and surfaces, tangent and normal vectors. Evaluation of line and surface integrals.
5. Conservative fields and their potentials. The divergence and Stokes's theorems.
6. Orthogonal curvilinear coordinates. Grad, div and curl in cylindrical and spherical polar coordinates

Teaching methods

 Delivery type Number Length hours Student hours Workshop 10 1.00 10.00 Lecture 33 1.00 33.00 Private study hours 107.00 Total Contact hours 43.00 Total hours (100hr per 10 credits) 150.00

Private study

- Studying notes and online media between lectures: 53 hours
- Doing problems: 40 hours
- Exam preparation: 14 hours

Opportunities for Formative Feedback

Regular problems sheets.

Methods of assessment

Coursework
 Assessment type Notes % of formal assessment Problem Sheet . 10.00 Online Assessment Online MCQ 5.00 Total percentage (Assessment Coursework) 15.00

There is no resit available for the coursework component of this module. If the module is failed, the coursework mark will be carried forward and added to the resit exam mark with the same weighting as listed above.

Exams
 Exam type Exam duration % of formal assessment Open Book exam 2 hr 30 mins 85.00 Total percentage (Assessment Exams) 85.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated