Vector & Complex Calculus - MAT00047I
Module summary
This module encompasses two subjects. Vector calculus is about differentiation and integration of scalar and vector fields in 2 and 3 dimensions. Complex analysis concerns the differentiation and integration of functions of a complex variable.
Related modules
Additional information
Pre-requisite modules:
- Foundations & Calculus
- Multivariable Calculus & Matrices
Module will run
| Occurrence | Teaching period |
|---|---|
| A | Semester 1 2025-26 |
Module aims
This module aims to deepen and extend knowledge from first year Calculus, with a more mature look at the fundamental concepts of infinitesimal calculus from the viewpoint of vector-valued functions of many variables. The three variable case 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.
The two variable case leads to the notion of differentiable functions of a complex variable which have very many properties beyond those of the differentiable functions of real variables that were studied previously. These underpin the important technique of contour integration, which is useful for evaluating real integrals.
Module learning outcomes
By the end of this module, students will be able to:
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Compute gradient, planar curl, divergence, curl, laplacian, dot product, and cross product, and work with algebraic and differential identities for these operations.
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Compute line integrals and flux integrals of vector fields. Compute the integral of a scalar field over a curve, surface, or solid region. Transform a triple integral using a given substitution, or apply cylindrical or spherical coordinates when appropriate.
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Apply the Fundamental Theorem of Line Integrals, Green’s theorem, Gauss’ theorem, and Stoke’s theorem.
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Apply the Cauchy-Riemann equations and discuss when a function is analytic.
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Compute the Laurent series and identify the poles of a complex function.
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Use the Residue Theorem to compute the value of a definite integral
Module content
In the first part of 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.
The second part of this module will extend the ideas developed about real functions to complex functions, develop the theory of holomorphic functions, and apply this theory to understand problems arising in real analysis or calculus. This lays the groundwork for future study of Fourier analysis which is widely used in real-world applications. These techniques are used by many graduates in their day-to-day work.
Indicative assessment
| Task | % of module mark |
|---|---|
| Closed/in-person Exam (Centrally scheduled) | 100 |
Special assessment rules
None
Additional assessment information
There will be five formative written assignments, with marked work returned in the seminars
Indicative reassessment
| Task | % of module mark |
|---|---|
| Closed/in-person Exam (Centrally scheduled) | 100 |
Module feedback
Current Department policy on feedback is available in the 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.
H A Priestley "Introduction to Complex Analysis" (2 ed) OUP 2003, JBM S7
A Pinkus and S Zafrany, Fourier Series and Integral Transforms (S 7.39 PIN)