In this chapter, we discuss and illustrate the tools of evaluating partnership work as part of a continuous improvement that you will need to make when building partnerships. As you engage with the chapter, you will learn what evaluation is and how to use evaluation to ensure that partnerships work for the benefit of children and their families. Throughout the chapter, we will explore different methods for collecting data and analysis, including various strategies that will inform the improvement and change necessary for good work in partnership-building.

In this chapter, we begin by examining the term ‘family’ and how it is defined in different contexts. As we examine these different definitions you will come to understand the complexity of ‘family’ and the diverse ways in which families can be defined. We then explore some of the structural and functional definitions of the family before moving towards examining some of the underlying assumptions made about families within wider society, including how these assumptions might position families in educational contexts. Through this exploration, some of your own underlying assumptions may be challenged as you come to understand the importance of educators and families working together to achieve the best educational outcomes for children. The chapter continues by discussing the idea of a subjective definition of families and what this might mean for you as an educator. We then move toward the term ’partnership’ and explore some of the barriers and opportunities to partnership work and how they can be harnessed and/or overcome. The chapter concludes by introducing the notion of innovative partnership work, your role in it as an educator, and the importance of this work in the educational context.

In this chapter, we begin by examining the role of reflective practice in partnership work and in doing so highlight the importance of knowing yourself and who you are as person. In Chapter 7, we critically examine reflective practice and how it unfolds in partnership work. In this chapter, we introduce the notion of reflective practice and how it prepares educators to begin thinking about how to come together with families. As you commence this journey of self-reflection, you will come to understand the complexity of partnership work and the skills you might need to develop to effectively engage with difference. We then explore some of the key ideas underpinning the planning of partnership work, including the importance of communication and open and positive mindsets, as well as the idea of active engagement and development of intercultural knowledge and capabilities. Through this examination, you will come to understand the first premise of the TWINE Model of Partnership so that you can identify as well as learn how to draw on this premise of the model when planning for partnership work.

In this chapter, we extend the learnings from Chapter 4 to expand your knowledge and skills on reflective practice for building effective and dynamic relationships for partnerships. You will understand how further elements of the TWINE Model of Partnership inform your reflective practice in partnership work. You will also come to learn about tools of reflective practice and how these tools can be useful in helping you to build meaningful relationships that contribute to partnerships with families and communities. This chapter will invite you to challenge yourself by asking key questions that will help you to become a reflective practitioner who builds dynamic relationships with children, families and communities.

In this chapter, we begin by examining the importance of trust in partnership work. We will then discuss the final premise of the TWINE Model of Partnership - to adapt. Through this premise, we will explore concepts such as participatory action, mapping out timelines, funding and resourcing a partnership. We will also examine some of the common challenges that might be faced in partnership work and discuss the ways these challenges might be overcome in practice.

Problems involving mass, momentum and energy transport in one spatial direction in a Cartesian co-ordinate system are considered in this chapter. The concentration, velocity or temperature fields, here denoted field variables, vary along one spatial direction and in time. The ‘forcing’ for the field variables could be due to internal sources of mass, momentum or energy, or due to the fluxes/stresses at boundaries which are planes perpendicular to the spatial co-ordinate. Though the dependence on one spatial co-ordinate and time appears a gross simplification of practical situations, the solution methods developed here are applicable for problems involving transport in multiple directions as well.

There are two steps in the solution procedure. The first step is a ‘shell balance’ to derive a differential equation for the field variables. The procedure, discussed in Section 4.1, is easily extended to multiple dimensions and more complex geometries. The second step is the solution of the differential equation subject to boundary and initial conditions. Steady problems are considered in Section 4.2, where the field variable does not depend on time, and the conservation equation is an ordinary differential equation. For unsteady problems, the equation is a partial differential equation involving one spatial dimension and time. There is no general procedure for solving a partial differential equation; the procedure depends on the configuration and the kind of forcing, and physical insight is necessary to solve the problem. The procedures for different geometries and kinds of forcing are explained in Sections 4.4–4.7.

The conservation equations in Sections 4.2 and 4.4–4.7 are linear differential equations in the field variable—that is, the equations contain the field variable to the first power in addition to inhomogeneous terms independent of the field variable. For the special case of multicomponent diffusion in Section 4.3, the equations are non-linear in the field variable. This is because the diffusion of a molecular species generates a flow velocity, which contributes to the flux of the species. The conservation equation for the simple case of diffusion in a binary mixture is derived in Section 4.3, and some simple applications are discussed.

In Section 4.8, correlations for the average fluxes presented in Chapter 2 are used in the spatial or time evolution equations for the field variables.

Convection can be neglected when the Peclet number is small, and the field variables are determined by solving a Poisson equation ∇‡2Φ fv + S = 0 or a Laplace equation ∇‡2Φ fv = 0, subject to boundary conditions, where Φfv and S are the field variable and the rate of production per unit volume, respectively. It is necessary to specify two boundary conditions in each co-ordinate to solve these equations. The separation of variables procedure is the general procedure to solve these problems in domains where the boundaries are surfaces of constant co-ordinate. This procedure was earlier used in Chapters 4 and 5 for unsteady one-dimensional transport problems.

The procedure for solving the heat conduction equation in Cartesian co-ordinates is illustrated in Section 8.1. The ‘spherical harmonic’ solution for the Laplace equation in spherical co-ordinates is derived using separation of variables in Section 8.2, first for an axisymmetric problem of the heat conduction in a composite, and then for a general three-dimensional configuration. There are two types of solutions, the ‘growing harmonics’ that increase proportional to a positive power of r, and the ‘decaying harmonics’ that decrease as a negative power of r, where r is distance from the origin in the spherical co-ordinate system.

An alternate interpretation of the decaying harmonic solutions of the Laplace equation as superpositions of point sources and sinks of heat is discussed in Section 8.3. It is shown that the each term in the spherical harmonic expansions is equivalent to a term obtained by the superposition of sources and sinks in a ‘multipole expansion’. A physical interpretation of the growing harmonics is also provided.

The solution for a point source is extended to a distributed source in Section 8.4 by dividing the distributed source into a large number of point sources and taking the continuum limit. The Green's function procedure for a finite domain is illustrated by using image sources to satisfy the boundary conditions at planar surfaces.

Cartesian Co-ordinates

Consider the heat conduction in a rectangular block of length L and height H, in which the temperature is T0 at x = 0 and x = L, TA at y = 0 and TB at y = H, as shown in Fig. 8.1.

Our work on differentiation in the previous chapter has brought us to a fork in the road. We can pursue the implications of the first fundamental theorem to obtain techniques for computing integrals. Alternately, we can use the Fermat and monotonicity theorems to further develop the relationship between functions and their derivatives, leading to new techniques of calculating limits, approximation of functions by polynomials, use of integration to measure arc length, surface area and volume, and error estimates for numerical calculations of integrals.We have chosen to take up integration in this chapter. If you are more interested in the other applications of differentiation you can read Chapter 6 first.

The Second Fundamental Theorem

A function F is called an anti-derivative of f if F = f . Let us make some observations regarding the existence and uniqueness of anti-derivatives:

• 1. Not every function has an anti-derivative. By Darboux's theorem (Theorem 4.5.12), if f = F then f has the intermediate value property. Thus, a function with a jump discontinuity, like the Heaviside step function or the greatest integer function, cannot have an anti-derivative.

• 2. On the other hand, the first fundamental theorem shows that every continuous function on an interval has an anti-derivative.

• 3. A function's anti-derivative is not unique. For example, both sin x and 1 + sin x are anti-derivatives of cos x.

• 4. On the other hand, two anti-derivatives of the same function over an interval can differ only by a constant. Theorem 4.5.7 states that if F = G on an interval I, then F G is constant. Thus, every anti-derivative of cos x over an interval I has to have the form sin x + C, where C ∊ R.

• 5. Over non-overlapping intervals, two anti-derivatives of a function need not differ by the same constant. For example, the Heaviside step function and the zero function are anti-derivatives of the zero function over (-∞,0) ∪ (0,∞).

The first fundamental theorem established a connection between integration and differentiation: if we are able to calculate the definite integrals of a continuous function, then the first fundamental theorem gives us its anti-derivative. The next theorem uses that connection to provide an approach for evaluating definite integrals by using anti-derivatives.

Calculus can be described as the study of how one quantity is affected by another, focusing on relationships that are smooth rather than erratic. This chapter sets up the basic language for describing quantities and the relationships between them. Quantities are represented by numbers and you would have seen different kinds of numbers: natural numbers, whole numbers, integers, rational numbers, real numbers, perhaps complex numbers. Of all these, real numbers provide the right setting for the techniques of calculus and so we begin by listing their properties and understanding what distinguishes them from other number systems. The key element here is the completeness axiom, without which calculus would lose its power.

The mathematical object that describes relationships is called “function.” We recall the definition of a function and then concentrate on functions that relate real numbers. Such functions are best visualized through their graphs, and this visualization is a key part of calculus. We make a small beginning with simple examples. A more thorough investigation of graphs can only be carried out after calculus has been developed to a certain level. Indeed, the more interesting functions, such as trigonometric functions, logarithms, and exponentials, require calculus for their very definition.

Field and Order Properties

We begin with a review of the set R of real numbers, which is also called the Euclidean line. It is a “review” in that we do not construct the set but just list its key attributes, and use them to derive others. For descriptions of how real numbers can be constructed from scratch, you can consult Hamilton and Landin [11], Mendelson [24], or most books on real analysis. The fundamental ideas underlying these constructions are easy to absorb, but the checking of details can be arduous. You would probably appreciate them more after reading this book.

What is the need for this review? Mainly, it is intended as a warm-up session before we begin calculus proper. Many intricate definitions and proofs lie in wait later, and we need to get ready for them by practising on easier material. If you are in a hurry and confident of your basic skills with numbers and proofs, you may skip ahead to the next section, although a patient reading of these few pages would also help in later encounters with linear algebra and abstract algebra.

The momentum flux or the force per unit area on a surface within a fluid can be separated into two components: the pressure and the shear stress. The latter is due to variations in the flow velocity, while the former is present even when there is no flow. Pressure has no analogue in mass and heat transfer, where the fluxes are entirely due to the variations in the concentration/temperature fields. The fluid pressure is the compressive force per unit area exerted on a surface within the fluid in the direction perpendicular to the surface. At a point within the fluid, the pressure is a scalar which is independent of the orientation of the surface; the direction of the force exerted due to the pressure is along the perpendicular to the surface.

There is a distinction between the thermodynamic pressure and the dynamical pressure that drives fluid flow. The thermodynamic pressure is an absolute pressure which is calculated, for example, using the ideal gas equation of state. In contrast, flow is driven by the pressure difference between two locations in an incompressible flow. The velocity field depends on the variations in the dynamical pressure, and the flow field is unchanged if a constant pressure is added everywhere in the domain for an incompressible flow.

A potential flow is a limiting case of a pressure-driven flow where viscous effects are neglected. Some applications of potential flows are first reviewed in Section 6.1. The velocity profile and the friction factor for the laminar flow in a pipe is derived in Section 6.2. As discussed in Chapter 2, there is a transition from a laminar to a turbulent flow when the Reynolds number exceeds a critical value. The salient features of a turbulent flow are discussed in Section 6.3. The oscillatory flow in a pipe due to a sinusoidal pressure variation across the ends is considered in Section 6.4. This flow is used to illustrate the use of complex variables for oscillatory flows, and the approximations and analytical techniques used in the convection-dominated and diffusion-dominated regimes.

Potential Flow: The Bernoulli Equation

At high Reynolds number, viscous effects are neglected in the bulk of the flow, and there is a balance between the pressure, inertial and body forces.

The decimal expansion of a real number is an instance of an infinite series. For example, 3.14159 … can be viewed as the sum of the series created by its digits: . The convergence of such a series is established by a comparison with the geometric series å¥n=0 1/10n. Isaac Newton realized that by replacing the powers of 1/10 with powers of a variable x, one can create real functions that can be easily manipulated in analogy with the rules of decimal expansions. He developed rules for their differentiation and integration as well as a method for expanding inverse functions in this manner. Today's historians believe that for Newton, calculus consisted of working with these “power series.” (See Stillwell [32, pp 167–70].) In this approach, the main task is to express a given function as a power series, after which it becomes trivial to perform the operations of calculus on it. In the first two sections of this chapter we shall study the general properties of power series, and then the problem of expressing a given function as a power series.

A century after Newton, Joseph Fourier replaced the powers xn with the trigonometric functions sinnx and cosnx to create new ways of describing functions. The “Fourier series” could model much wilder behavior than power series, and forced mathematicians to revisit their notions of what is a function, and especially the definition of integration.We give a brief introduction to this topic in the third section.

In our final section, we introduce sequences and series of complex numbers. These bring further clarity to power and Fourier series, and even unify them through the famed identity of Euler: eix = cos x + i sin x.

Power Series

Let us recall our study of Taylor polynomials in §6.3. Given a function f that can be differentiated n times at x = a, we define the Taylor polynomial

Tn is intended to be an approximation for f near x = a, with the hope that the approximation improves when we increase n. These hopes are not always realized, but in many cases they are. (The remainder theorem gives us a way to assess them.) It is natural to make the jump from polynomials to series, and consider the expression

The questions that arise here are: (a) For which x does this series converge, and (b) When it converges, does it sum to f (x)? To tackle these questions, we initiate a general study of series of this form.