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Basic algebra and arithmetic  

This chapter provides an overview of basic algebra and arithmetic. To avoid ambiguities, the text introduces an order ranking the priorities of the various basic arithmetical operations has been agreed; this convention , which can be remembered by the acronym BODMAS. According to BODMAS, things occurring inside a bracket must be evaluated first; the next most important operation is 'orders', as in powers and roots; then, it is division or multiplication, as they appear left-to-right;, and lastly addition or subtraction. The chapter also then looks at powers, roots, and logarithms; quadratic equations; simultaneous equations; the binomial expansion; arithmetic and geometric progressions; and partial fractions. The situations in which partial fractions arise involve the ratios of two polynomials, where the denominator can be written as the product of simpler components.

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Cover Foundations of Science Mathematics

Complex numbers  

This chapter assesses complex numbers. It explains that Iif any number, integer or fraction, positive or negative, is multiplied by itself, then the result is always greater than, or equal to, zero. Although the construct of imaginary numbers appears arbitrary, complex numbers turn out to be very useful for tackling many real-life problems. The chapter then looks at basic algebra and complex numbers, before considering the Argand diagram, the imaginary exponential, and roots and logarithms. It also studies de Moivre's theorem and hyperbolic functions. The hyperbolic functions are related to the sines, cosines, and so on, of imaginary angles. The derivatives, and integrals, of hyperbolic functions are also readily ascertained by exploiting the convenient differential properties of the exponential function.

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Cover Foundations of Science Mathematics

Curves and graphs  

This chapter discusses curves and graphs. The relationship between two quantities, generically called x and y, is often best understood when displayed in the form of a graph. That is when linked pairs of x and y values are plotted as distinct points on a piece of paper, and joined together to form a curve. The chapter first looks at straight lines before considering parabolas. Both the straight line and the parabola are special cases of a family of curves known as polynomials. The similarity between them is more obvious from the algebraic form of the equation that defines them. Then, the chapter then studies powers, roots, and logarithms; circles; and ellipses.

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Data analysis  

This chapter focuses on data analysis. One of the things that sets science apart from other systems of trying to make sense of the world is the use of empirical evidence. A scientific theory must therefore be able to make predictions that can be compared with observations. Since the amount of data obtained in an experiment is limited, and subject to noise in the measurement process and uncertainty in the parameters required for their analysis, the conclusions reached have to be couched in conditional terms with an appropriate statement about their reliability. The chapter then looks at the basic rules of drawing inferences, common probability distributions, parameter estimation, propagation of errors, and model comparison.

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Differentiation  

This chapter investigates differentiation, starting with a discussion of gradients and derivatives. While the intersections of the curve with the x and y axes may be of interest, it is often more important to know the slope at any given point; that is, how quickly y increases, or decreases, as x changes, and vice versa. The chapter notes that Tthis issue is at the heart of the topic of differentiation. It also highlights that Oone of the most fundamental properties of derivatives is their linearity. The chapter thenThen, the chapter looks at exponentials and logarithms, products and quotients, functions of functions, and the concepts of maxima and minima. It also compares between implicit and logarithmic differentiation.

Book

Cover Foundations of Science Mathematics

D.S. Sivia, J.L. Rhodes, and S.G. Rawlings

Foundations of Science Mathematics bridges the gap between school and university, and spans covers a large range of topics, from basic arithmetic and algebra to calculus, Fourier transforms, and elementary data analysis. Problems and worked solutions are presented in an informal and readable tutorial style, making the text accessible to the novice, while its concise nature ensures that it is a useful reference for the experienced professional. Mathematics plays a key part in every quantitative and theoretical study, and is taught to all science and engineering students to varying degrees at college and university level. This text spans a large range of topics, from basic arithmetic and algebra to calculus and Fourier transforms and bridges the gap between school and university. Other topics covered included are curves, graphs, differentiation, integration, trigonometry, vectors, matrices, linear integrals, data analysis, and Taylor series.

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Fourier series and transforms  

This chapter examines Fourier series and transforms. Previous chapters show how Taylor series could be used to mimic an arbitrary function by a simple low-order polynomial in the neighbourhood of a particular point. An alternative approximation is provided by a Fourier series, which is appropriate when the curve of interest is periodic. The general characteristics of the Fourier series, with its high-frequency oscillatory behaviour when sharp features are involved, means that a term-by-term differentiation is an extremely ill-advised operation. The chapter then looks at the Fourier integral. It also considers the auto-correlation function of Fourier transforms, which provides information on the distribution of the distances between various parts of the structures in f(x).

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Integration  

This chapter evaluates integration, starting with a discussion of areas and integrals. It explains that Itntegration deals with the 'area under a curve', which; this is intimately linked with the important questions of the average and cumulative behaviour of y, over some range in x. Although an integral is defined as the limiting form of a summation, it is usually evaluated by noting that 'integration is the reverse of differentiation'. The chapter clarifies that, Jjust as for the case of derivatives, one of the most fundamental properties of integrals is their linearity. The chapterIt then looks at inspection and substitution, partial fractions, integration by parts, and the reduction formula. Finally, the chapter examines symmetry, tables, and numerical integration.

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Line integrals  

This chapter studies line integrals. It notes that the result of a line integral is, in general, dependent on the path followed from start to finish. The exception to this is when the integrand is an exact differential, and then the line integral is independent of the route taken and a complicated path can be replaced by a simpler one to facilitate the calculation. The chapter also explains that exact differentials are related to conservative fields (or forces) and state functions. For example, the change in the gravitational potential energy of an object depends only on the difference in its height and not on any other characteristics of the motion. Similarly, in thermodynamics, the value of a function-of-state depends literally on the state (temperature, pressure, volume, etc.) of the system and not on how it got there.

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Matrices  

This chapter describes matrices. The simplest definition of a matrix is that it is a rectangular array of numbers, usually enclosed in large round brackets. Depending on their size (or shape), and the properties of their elements, matrices are often given qualifying names. Two properties of matrices which are frequently met in practice is that they are real and symmetric. The former means that the elements are not complex numbers; the second, which implicitly assumes a square matrix, means that the matrix remains the same if its rows and columns are interchanged. The chapter then looks at matrix arithmetic, determinants of matrices, inverse matrices, linear simultaneous equations, transformations, eigenvalues and eigenvectors, and diagonalization.

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Multiple integrals  

This chapter highlights multiple integrals. It begins by presenting a couple of physical examples of multiple integrals, including double integrals which can also be called surface integrals. As a concrete example of how to calculate multiple integrals, the chapter considers a very easy case, namely, working out the formula for the area of a right-angled triangle. This illustrates that (for a well-behaved function) the order of integration in a multiple integral does not matter, and that it can be interchanged. The chapter then looks at the choice of coordinates. Multiple integrals, like many other mathematical operations, become much simpler if they are formulated in a coordinate system which matches the geometry of the problem.

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Ordinary differential equations  

This chapter reviews ordinary differential equations (ODEs), which involve only normal differentials of y(x), that is, dy/dx and its derivatives. The order of a differential equation is the highest derivative of y that appears in it. When applied to differential equations, the concept of linearity means that y and its derivatives are raised to no power higher than the first. Equations where this is not true are called non-linear. The degree of a differential equation is the power to which the highest derivative (of y) is raised, but the chapter only deals with those of the first. The chapter begins by looking at first-order linear ODEs before considering second-order linear ODEs.

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Partial differential equations  

This chapter discusses partial differential equations (PDEs). It begins by presenting elementary cases of PDEs, which highlights that PDEs give rise to 'functions of integration', in contrast to ordinary differential equations (ODEs), which have 'constants of integration'. The chapter then focuses on one method of solving PDEs: the separation of variables. The power of the technique lies in its ability to reduce a PDE to an equivalent set of ODEs. On a practical note, the choice of the constant used and the subsequent order in which the boundary conditions are implemented plays an important part in facilitating a successful outcome. In this context, the recommended preliminary diagram is helpful because it suggests that the solution of the PDE will be a decaying function in the x direction and an oscillatory one in y.

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Partial differentiation  

This chapter explores partial differentiation. It focuses principally on the case where a parameter z is a function of just two variables, x and y: z = f(x, y). We can visualize such a situation by thinking of z as representing a height, with x and y being the two-dimensional coordinates of a point on the (flat) ground. While a realistic rendering of the topographical scene requires artistic ability, a more practical means of displaying the salient information is to draw a contour map; these are found in geographical atlases, where hills and valleys are depicted by a series of lines which join together regions of equal altitude. The chapter then looks at second and higher derivatives, increments and chain rules, the Taylor series, maxima and minima, and constrained optimization.

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Taylor series  

This chapter focuses on the Taylor series. It points out that, Wwhen dealing with a complicated function, it can be useful to approximate the functionit with one of a simpler form. While the latter may not represent a complete and accurate description of the situation at hand, it frequently provides the only means of making analytical progress. There are many approximations that could be used, of course, but it is the one that captures the salient features that is most helpful. The chapter then looks at derivation of the Taylor series before providing some common examples. It also considers the radius of convergence, l’Hôpital’s rule, and the Newton-Raphson algorithm.

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Trigonometry  

This chapter examines trigonometry. It starts with a discussion of angles and circular measure. An angle is a measure of rotation or turn, and it is usually specified in degrees. The chapter notes that, despite our familiarity with degrees, a circular measure that is often more useful in a mathematical context is a radian. Unless degrees are mentioned explicitly, it is best to assume that all angles are implicitly given in radians. The chapter then looks at sines, cosines, and tangents; Pythagorean identities; compound angles; factor formulae; inverse trigonometric functions; and the sine and cosine rules. The sine and cosine formulae relate the lengths of the sides of a general triangle to its angles.

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Vectors  

This chapter addresses vectors, which are defined by both a magnitude (or modulus), a positive number, normally with an associated unit, and a direction in space. Many physical quantities, such as position and velocity, are vectors which must be manipulated in slightly different ways to scalars like mass, length, and time. The chapter then looks at the Cartesian coordinate system, before considering the addition, subtraction, and scalar multiplication of vectors. It also studies scalar and vector products, as well as scalar triple and vector triple products. Finally, the chapter discusses polar coordinates, examining the two commonly used generalizations of the polar formulation when working in three dimensions.