A primary source of excitation in marine dynamics is the ocean environment, which is often characterized as a random process. Therefore, objective analysis of resulting dynamics is presented in terms of averages, or probabilities. For example, it is possible to determine, within the limits of the modeling assumptions, the average of the l/3 largest waves, or the average of the 1/1000 bow accelerations. The basis for these averages is linear theory and the Fourier transform. This chapter shows how the frequency decomposition of a time series can be achieved by Fourier analysis, resulting in a “mean square density” spectral density function. The assumption that the process is stationary and ergodic results in temporal statistics, e.g. process mean and mean square, are equal to ensemble statistics. Therefore a single time series record may be used to estimate probability density functions and statistical properties. Probability density functions (PDF) for the elevations (Gaussian) and amplitudes (Rayleigh, if the process is narrow banded) are given. Extreme value PDF’s and most probable maxima relations are derived allowing for the estimate of the largest response in N encounters.

This is a discussion of sequences and first-order recurrence (or difference) equations and the behaviour of the solutions to such equations. It contains some economic applications.

We give a discussion of some basic mathematical concepts and terminologies connected with sets, numbers, functions and graphs.

We start by introducing the key ingredients in macroeconomic modelling: investment, production, income and consumption, and explain the corresponding equilibrium conditions. Modelling these quantities in discrete time, we describe the multiplier-accelerator model, a classic model of macroeconomic dynamics, and an example of a second-order recurrence equation. We then embark on describing how to solve linear constant-coefficient second-order recurrence equations in general. The general solution is the sum of the solution of a corresponding homogeneous equation and a particular solution. There is a general method for determining the solution of the homogeneous equation, involving the solution of a corresponding quadratic equation known as the auxiliary equation.

A reduced order model for marine vehicle dynamics is the simple linear spring-mass-damper system. However, the various terms in the equation of motion differ in detail from their mechanical counterparts. The usual balance between mechanical inertial, damping, and stiffness loads with external forcing is maintained, but now includes additional effects reflecting the presence of the fluid. Individual coefficient matrices correspond to the mass of the platform plus the mass of the water being accelerated; the linear damping coefficient of the system due to viscous effects and the generation of radiating waves due to platform motion; a linear restoring force/moment coefficient due to hydrostatic pressure and/or mooring lines; and an external exciting force/moment due to incident waves, wind, tow lines, etc. Ideal fluid theory is introduced to model the hydrodynamic forces implicit in the marine system’s equations of motion. The purpose is not to give a detailed derivation of basic hydrodynamics, but rather to describe the assumptions necessary to apply the useful ideal, potential theory and understanding when the theory will be successful and, equally important, when it will not.

It is explained what is meant by a function defined implicitly and how the derivative of an implicitly defined function can be determined via partial differentiation. The general concept of the contour of a two-variable function is presented, together with the special case of this when the function is a production function, and the contours are known as isoquants. It is explained how the slopes of contours can be determined. Then, the concept of homogeneous functions and the connected economic interpretation of returns to scale are considered, along with Euler's Theorem and its economic interpretation in terms of marginal product of labour and marginal product of capital.

Basic concepts in finance are introduced and modelled via first-order recurrence equations. In particular, we discuss compound interest, present value and the present value of an annuity.

The concept of consumer surplus is introduced and this motivates the problem of determining the area under the graph of a function. We indicate the connection between this problem and anti-derivatives (or integrals), defining what we mean by a definite integral. We illustrate with some examples after developing a repertoire of standard integrals.

This chapter introduces the important idea of a vector through the example of bundles of goods. The dot product of two vectors is defined and it is shown how a budget constraint can be expressed in terms of dot product. It is explained how, in order to rank bundles according to a particular consumer's preference, we can use a utility function. Indifference curves are defined as the contours of the utility function. Linear and convex combinations and the concept of a convex set are explained. The utility maximisation problem -- to maximise utility subject to a budget constraint -- is explored and the relevance of convexity is emphasised.

Regarding the key components of macroeconomics as continuous (rather than discrete) and considering the corresponding dynamics leads to differential equations. In this chapter, we focus on first-order differential equations and consider two methods (applicable to certain types of equation): separation and the use of integrating factors. We look at economic applications to continuous-time price adjustment and continuous cash flows.

A mathematical discrete-time population model is presented, which leads to a system of two interlinked, or coupled, recurrence equations. We then turn to the general issue of how to solve such systems. One approach is to reduce the two coupled equations to a single second-order equation and solve using the techniques already developed, but there is another more sophisticated way. To this end, we introduce eigenvalues and eigenvectors, show how to find them and explain how they can be used to diagonalise a matrix.