Multivariable Calculus

• Credit value: 15 credits at Level 5
• Convenor: Dr Richard Pymar
• Assessment: a problem set (50%) and a two-hour examination (50%)

In this module we extend fundamental calculus concepts to handle functions of multiple variables, providing essential mathematical tools for analysing complex systems and optimisation problems. You will develop proficiency in partial differentiation, multiple integration and differential equations.

Indicative syllabus

• Functions and differentiation: functions, limits, continuity, differentiation, functions of two variables, partial differentiation, vector products, tangent planes, directional derivatives, multivariate functions
• Applications of partial differentiation: higher derivatives, stationary points, local vs global extrema, Lagrange multipliers, Lagrange multipliers with multiple constraints, the chain rule, Taylor polynomials
• Integration: integration of two-variable functions, integrals with infinite limits, double integrals for rectangular regions, non-rectangular regions of integration, changing the order of integration, unbounded regions of integration, change of variables, polar coordinates
• Differential equations: terminology, separation of variables, exact 1st-order ODEs, integrating factors, homogeneous ODEs, special families of ODEs

By the end of this module you will be able to:

• understand and use mathematical and statistical techniques
• construct mathematical arguments to establish a range of mathematical results
• understand the processes and limitations of mathematical approximation and computational mathematics.