Aims: To introduce and apply methods and techniques from multivariable calculus, focusing on two and three dimensions.
Content: This module will introduce you to a range of definitions, methods and techniques in the field of multivariable calculus. We will consider the concepts of parametrised curves and surfaces, define functions of several variables, and learn about different types of derivatives for these functions. We will also learn how to integrate such functions along curves, and over surfaces and volumes in two and three dimensions. The ideas we cover are the fundamental basis for a huge range of scientific theories, ranging from flows of fluid, traffic and pedestrians to the motion of galaxies, the transfer of heat, and the propagation of sound and water waves. They are also of fundamental use in a range of areas of pure mathematics too, including in differential geometry, the study of partial differential equations and dynamical systems.
Objectives: By the end of this module, students should be able to:
- Define the concept of scalar-valued and vector-valued functions of one or more variables
- Interpret and provide parametric representations of curves and surfaces
- Perform coordinate transformations for multivariable functions
- Define and compute different notions of differentiation for functions from R^n to R^m, including partial derivatives, the gradient, Jacobian matrix, directional derivative, divergence and curl
- Explain the definition of the Riemann integral of a multivariable function and provide geometric interpretations of such integrals
- Demonstrate understanding of integral theorems relating line, surface and volume integrals and use these theorems to evaluate such integrals
- Compute integrals of functions over simple domains and justify the change of area and volume elements when converting coordinates.
Books: For a list of recommended books, see the reading list on Talis.