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Vector Calculus - MAT00030I

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• Department: Mathematics
• Credit value: 10 credits
• Credit level: I
• Academic year of delivery: 2022-23

Module will run

Occurrence	Teaching period
A	Autumn Term 2022-23

Module aims

The Core modules in the second year of mathematics programmes cover material which is essential for accessing a wide range of topics later on. Such material underpins the development of mathematics across all three of the major streams we offer through our single-subject degrees, and it is also very important for many combined programmes. As well as covering fundamental material, these modules address applications and techniques which all students will subsequently be able to draw on in various contexts.

As part of the broad aims outlined above, in this module we deepen and extend the Stage 1 Calculus module, with a more mature look at the fundamental concepts of infinitesimal calculus from the viewpoint of vector-valued functions of many variables. This is the mathematics which underpins all continuous processes in the three space dimensions of our world, and which is therefore essential to the application of mathematics in the natural sciences.

Module learning outcomes

Subject content

In this module, we discuss the three essential ingredients of calculus—continuity, differentiability, and integrability—bringing out the distinctive flavour of each theory, and describing their inter-relationships and applications; we gain a deeper understanding of the three famous differential operators of classical vector calculus: div, grad and curl; we describe the theorems of Stokes and Gauss which link these topics together.

Syllabus:

• Functions from R^n to R^m: total derivative, Jacobian, chain rule

• Curves; line integrals; vector fields; gradient, potential and fundamental theorem

• Planar integrals; change of variables; Green's theorem

• Surface and volume integrals; spherical and cylindrical coordinates; change of variables

• Grad, div and curl and associated identities; index notation and summation convention

• Gauss's Theorem, Stokes's Theorem and examples

 

Academic and graduate skills

• Our second year Core modules continue to develop themes which start in the first year. Vector Calculus develops the 3D calculus necessary to describe real-world spatially-varying continuous processes and provides powerful and widely-used tools and calculational techniques, thereby enabling the mathematical description of many processes in the natural sciences.

Indicative assessment

Task	% of module mark
Closed/in-person Exam (Centrally scheduled)	100

Special assessment rules

None

Indicative reassessment

Task	% of module mark
Closed/in-person Exam (Centrally scheduled)	100

Module feedback

Current Department policy on feedback is available in the undergraduate student handbook. Coursework and examinations will be marked and returned in accordance with this policy.

Indicative reading

• KF Riley, MP Hobson, SJ Bence, Mathematical Methods for Physics and Engineering: A Comprehensive Guide, CUP 2006 (3rd ed)
• H F Davis & A D Snider, Vector Analysis, Allyn & Bacon.