This chapter focuses on social individual differences in relation to second language learning. It explores how the social, cultural, and political context that a learner is situated in affects their success of second language learning. The chapter begins by explaining how society and social interaction that second language learners encounter influence the access they have to second language education. This includes the differences between foreign vs. second language learning contexts. It then focuses on social identity theory, acculturation theory relevant to immigrant learners, and transdisciplinary framework (by Douglas Fir Group). The chapter covers other socially constructed individual differences related to intersectionality, diversity–equity–inclusion (DEI), and heritage language learning. The chapter also addresses socially constructed biases related to race and ethnicity, social class, sexual orientation, and LGBTQ+. The chapter ends with a series of pedagogical recommendations that mitigate the impacts of socially constructed biases on second language learning in the classroom.

The cobweb model is introduced and analysed fully for the case of linear supply and demand, and conditions for stability are derived.

The inverse of a square matrix is defined and it is explained how the existence of an inverse is related to the question of whether systems of linear equations corresponding to the matrix have unique solutions. A method for determining when a matrix is invertible and, when it is, finding the inverse, is shown: this involves row operations. An explicit formula for the inverse in the 2 × 2 case is given. This is applied to IS--LM analysis, a macroeconomic application.

The profit-maximisation problem (for production of one good) is introduced as motivation for the development of general optimisation techniques. The general concept of a critical (or stationary) point is presented, together with the method for finding such points and classifying their nature in two different ways: by examining the sign of the derivative around the point and by using the second-derivative test. Optimisation on intervals and infinite intervals is then discussed (where the end-points must be taken into consideration). Additional economic and financial applications are given.

Matrices are introduced and it is explained how matrix addition, scalar multiplication and multiplication of two matrices works. As an example of matrix multiplication, it is demonstrated how investment portfolios can be modelled, and their returns in various states quantified by multiplication with a returns matrix. The concept of an arbitrage portfolio is explained.

This chapter introduces (with production function as an example) functions of more than one variable. Then we define partial and second partial derivatives and explain how to calculate them, and present the chain rule for partial differentiation.

Important special functions and their properties are described. In particular, the exponential function and its connection to continuous compounding is discussed, together with the logarithm and trigonometrical functions. It is explained how one can interpret not just integer, but rational and then irrational powers of positive real numbers. The derivatives of exponential, logarithmic and trigonometrical functions are studied.

The standard integrals of the previous chapter are of fairly limited use, so this chapter develops some much more widely applicable techniques. These are integration by substitution, integration by parts and integration by partial fractions.

Hydroelastic problems involve dynamically coupled, structurally elastic, hydrodynamic systems. The fluid can have many effects such as added inertia, additive hydrostatic stiffness, increased system damping, or external excitation (e.g. wave impact, variable current forces, etc.). This chapter illustrates some of the aspects of hydroelastic problems by deriving fundamental relationships and discussing a specific example - ship springing. Springing vibration is differentiated from whipping vibration by the source of excitation. Springing is excited by synchronous matching of the natural frequency with the incident wave encounter frequency while whipping is transient vibrations due to impact/slam loads. A well-developed energy method - the Rayleigh-Ritz method - is applied in the determination of fluid-structure resonance. For general marine vibrations, energy methods may be used when free surface effects are small or negligible. Fluid inertia effects are calculated using strip theory and Lewis form coefficients. Limitations of strip theory are discussed. A spherical globe mounted on a flexible pole submersed in water is given as an example of a hydroelastic system.

This chapter overviews broad issues related to language acquisition research and answers fundamental questions related to first language acquisition, such as What is language? and How do children learn their first language? It begins by introducing some components of language, such as grammar, vocabulary, pronunciation, pragmatics, and discourse. It also discusses language varieties (e.g., American English vs. Indian English) and explains their legitimacy. Then, the chapter addresses language learning during the first years of a child’s life. It also discusses bilingualism, especially of those who start learning multiple languages in their early years (simultaneous bilingualism). In order to understand second language acquisition for the rest of the textbook, the chapter focuses on a question: How are children able to learn a language without formal instruction? In order to answer this question, multiple language acquisition theories will be discussed.

This chapter focuses on instructed second language acquisition research that examines second language learning specifically in instructional contexts, broadly construed (e.g., classrooms, online learning, self-study). The scope of the research field is discussed by distinguishing it from broader second language acquisition research. In essence, the chapter addresses the primary question that researchers and teachers are interested in: Can instruction help learners develop their second language proficiency? The chapter then answers the follow-up question which is: Which types of instruction are more or less helpful? In answering this question, the chapter considers the goals that learners and other stakeholders, such as teachers and parents, have for their second language development. It discusses different types of second language knowledge (implicit and explicit knowledge) as the goal of second language instruction. Finally, the chapter explores pedagogical issues and ends by considering a taxonomy that describes various approaches and methodologies to second language teaching.

The general matrix formulation of a system of linear equations is described. It is explained that a system may have no, one or infinitely many solutions. We begin to describe a general approach to solving such systems by performing row operations on the augmented matrix in order to reduce this to echelon form. This chapter gives examples in the case where the system has a unique solution. (The next chapter considers other cases.)

The analysis in this chapter of marine platform motions is directly applicable to any floating system such as ships, offshore platforms, floating wind turbines, or wave energy devices. The basic underlying model is the classic linear spring-mass-damper system. The mass will be augmented by the added mass of the fluid; the damping will be the result of the dissipation of energy by waves; the linear spring will be due to hydrostatic effects plus any external stiffness such as mooring lines; and the exciting forces are due to incident waves. Depending on the body shape and mass distribution, the equations of motion can be dynamically/statically coupled. Wave excitation is comprised of Froude-Krylov and diffraction components. Solutions to the equations of motion in the frequency domain are expressed as RAO’s. The RAO is a linear operator representing the dynamic response of a system (e.g. displacement, acceleration, bending moment, etc.) per unit input, typically the incident wave amplitude. Once the rigid body dynamics are expressed as RAO’s, other quantities or dynamics of interest may be determined, e.g. relative motion, dynamic bending and shear.

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.