Vector Calculus is the natural progression of the Calculus of Several Variables (MA2001). Extending the concepts of differentiation and integration to vector-valued functions leads to many powerful and useful tools such as the famous div, grad and curl operators. These concepts describe compression, expansion, slope and swirling mathematically and therefore have important applications both within mathematics and beyond. The skills and knowledge you will learn on this course are a foundation for further study into differential geometry, fluid dynamics and continuum mechanics, to name but a few examples.

Pre-requisite modules: MA2001;

Recommended: MA1001 , MA1301 

• Understand the origins of integral methods and differential operators, as applied to vector and scalar fields.
• Appreciate the wide range of real-world applications and the physical basis of these tools, including in future study of areas such as: Fluid Dynamics, Continuum Mechanics, Differential Geometry and Topology.
• Evaluate line, surface and volume integrals of both vector and scalar fields.
• Calculate the divergence and curl of a vector field and the gradient and directional derivative of a scalar field.
• Switch between operator and integral methods using integral theorems.

You will be guided through learning activities appropriate to your module, which may include:

• Weekly face to face classes (e.g. labs, lectures, exercise classes)
• Electronic resources to support the learning (e.g. videos, exercise sheets, lecture notes, quizzes)

Students are also expected to undertake at least 50 hours of self-guided study throughout the duration of the module, including preparation of formative assessments.

Skills:
The capability to analyse and describe a range of properties of vector and scalar fields, including their derivatives and integrals.

Transferable Skills:
Calculation, visualisation and interpretation skills from vector calculus provide essential ingredients for many areas of mathematical modelling, as well as being of fundamental mathematical importance..

The course will be broken down into 7 topics:

1. Recap - “Calculus & vectors”: Partial differentiation, Integrals of more than one variable, Vector algebra, Vector differentiation
2. Vector fields, the del operator & the gradient - “The vector of calculus”: The directional derivative and the gradient vector field
3. Line integrals - “The forced path”: Parametric curves & their properties, The length of parametric curves, Natural parametrisation, Line integrals of scalar and vector fields, The fundamental theorem for line integrals, Conservative vector fields
4. The curl operator - “The calculus of swirling”: The continuous variation of circulation, The curl vector field and curl operator, Rotation and vorticity, Curl’s relation to conservative vector fields
5. Surface integrals & volume integrals - “Feeling the pressure”: Parametric surfaces and their areas, Scalar and vector integrals along surfaces, Volume integrals, Polar coordinates
6. The divergence operator - “The source and the sink”: Converging and diverging flow, The continuous variation of flux, The divergence operator, The Laplacian   
7. Integral theorems - “Connecting the dots”: Green’s theorem, The divergence theorem, Stokes’ theorem