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Department of Mathematical Sciences

# MATH2031 Analysis in Many Variables II

The description of most natural phenomena is based on models that involve functions of several variables. The highlight of this module is the study of two prominent partial differential equations, namely Laplace's equation and the heat diffusion equation. These equations relate two or more partial derivatives of an unknown function of several variables, and play an enormous role in science. The methods to solve these partial differential equations are cornerstones of applied mathematics.

### Term 1

The module deals with functions depending on n real variables. Whilst the mathematical properties and procedures are natural extensions of those for the one-variable case you are already familiar with, you will see that the notion of differentiability for a function of more than one variable is more subtle. We then move on to study ordered triples of functions of 3 real variables and their generalisations, which are called vector functions or vector fields. Vector algebra is so prodigiously rich in applications that it plays a crucial role in many areas of science. But vector calculus goes much beyond vector algebra. Differential and integral vector calculus opens the door to three great integration theorems: Green's theorem, Gauss's theorem (commonly known as the divergence theorem) and Stokes's theorem. All three theorems can be cast in the same general form: an integral over a region S is equal to a related integral over the boundary of S. Vector calculus was actually invented to provide an elegant formulation of the laws of physics but it has applications in many other mathematical contexts.

### Term 2

The second term concentrates on a number of mathematical methods for solving linear partial differential equations, subject to various boundary conditions. The method of separation of variables is explored, and the use of Fourier series in obtaining exact solutions is demonstrated. This approach is then generalised through the development of Sturm-Liouville Theory. Finally, we look at the solution of partial differential equations through Fourier (integral) transforms , with the Heat equation as the primary example, and generalised functions are introduced.

## Outline of Course

### Term 1

• Vector Algebra: Suffix notation and use for scalar and vector products.
• Functions on ℝn: Open subsets of ℝn, continuity and differentiability, gradient. Continuous partials imply differentiability.
• Local and global extrema of functions of more than one variable. Lagrange multipliers.
• Functions from ℝn to ℝm: Differentiation, chain rule, inverse and implicit function theorems. Curves given by f(x,y)=c, tangent planes and normal lines to surfaces. Local diffeomorphisms, orientation and relation with Jacobian.
• Vector Calculus and Integral Theorems: Div, curl, Laplacian (in Cartesian coordinates). Multiple integration. Line, surface and volume integrals. Change of variables. Stokes and divergence theorems. Conservative fields and scalar potential.

### Term 2

• PDEs: Discussion of Laplace, Poisson, Heat, Wave equations, origin and types of boundary and initial conditions. Examples of uniqueness, and solution by separation of variables. Use of Fourier series.
• Sturm-Liouville Theory: Generalised Fourier Series.
• Fourier Integrals: Solution of Heat equation on ℝn. Generalised Functions.

### Prerequisites

For details of prerequisites, corequisites, excluded combinations, teaching methods, and assessment details, please see the Faculty Handbook.