MMath Mathematics and Statistics

This unit introduce students to the calculus of functions depending on more than one variable, while emphasising its usefulness in physical applications.

The unit aims to introduce students to multivariable calculus, without assuming prior knowledge of linear algebra, while emphasising its usefulness in physical applications.
  

On the successful completion of the course, students will be able to:  

• Differentiate scalar and vector fields and physically interpret the associated operators; grad, curl and div. Determine whether a given vector field is conservative or solenoidal.   
• Construct, evaluate and interpret definite integrals of functions of two and three variables.  
• Verify specific examples of the classic integral theorem(s) of vector calculus in three dimensions; Green’s theorem, the divergence theorem and Stokes’ theorem.
• Evaluate vector products in two and three-dimensions. Evaluate square-matrix and vector products. Compute determinants of 2x2 and 3x3 real matrices. 
• Evaluate and manipulate functions of more than one variable. Sketch contours of functions of two variables near critical points. Sketch two-dimensional vector fields. Construct, evaluate and interpret partial derivatives of scalar and vector-valued functions of one, two and three variables.  Find and classify critical points of functions of two variables.

This course will be split as follows

Introduction
The differences between a scalar and vector quantity. The idea of a “matrix” and how we can define an algebra involving scalars/vectors and matrices. 

Scalar fields
These are functions that output a scalar value but can depend on multiple scalar values as “inputs”. For example f(x,y) rather than a function of a single variable f(x). Here we will discuss how to extend your existing understanding of differential calculus to these new multivariable functions. Having discussed how to differentiate such functions we move on to how we can introduce integrals of these same functions.  

Vector fields
In real-world problems we often deal with physical quantities best represented as a vector, this leads us to functions of multiple variables whose “output” is a vector. Again we introduce the ideas of differential/integral calculus in this new context ("vector calculus"). There are some famous theorems that we cover and spend some time examining them in a range of example problems.

Applications and examples
Finally we return to some of the key ideas in the course in the context of some physical applications. 

There are supervisions in alternate weeks which provide an opportunity for students' work to be marked and discussed and to provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.