Covers differentiation and integration, higher derivatives, partial derivatives, series expansion, integral transforms, convolution integrals, Laplace transforms, linear and time-invariant systems, linear ordinary differential equations, periodic functions, Fourier series and transforms, and matrix algebra.

This chapter introduces the continuous-time Fourier transform (CTFT) and its properties. Many examples are presented to illustrate the properties. The inverse CTFT is derived. As one example of its application, the impulse response of the ideal lowpass filter is obtained. The derivative properties of the CTFT are used to derive many Fourier transform pairs. One result is that the normalized Gaussian signal is its own Fourier transform, and constitutes an eigenfunction of the Fourier transform operator. Many such eigenfunctions are presented. The relation between the smoothness of a signal in the time domain and its decay rate in the frequency domain is studied. Smooth signals have rapidly decaying Fourier transforms. Spline signals are introduced, which have provable smoothness properties in the time domain. For causal signals it is proved that the real and imaginary parts of the CTFT are related to each other. This is called the Hilbert transform, Poisson’’s transform, or the Kramers–Kronig transform. It is also shown that Mother Nature “computes” a Fourier transform when a plane wave is propagating across an aperture and impinging on a distant screen – a well-known result in optics, crystallography, and quantum physics.

In this chapter, the aim is to visualize wave dynamics in one dimension as dictated by the Schrödinger equation. The necessary numerical tools are introduced in the first part of the chapter. Via discretization, the wave function is represented as a column vector and the Hamiltonian, which enters into the Schrödinger equation, as a square matrix. It is also seen how different approximations behave as the numerical wave function reaches the numerical boundary – where artefacts appear. This numerical framework is first used to see how a Gaussian wave packet would change its width in time and, eventually, spread out. Two waves interfering is also simulated. And wave packets are sent towards barriers to see how they bounce back or, possibly, tunnel through to the other side. In the last part of the chapter, it is explained how quantum measurements provide eigenvalues as answers – for any observable physical quantity. This, in turn, is related to what is called the collapse of the wave function. It is also discussed how a quantity whose operator commutes with the Hamiltonian is conserved in time. Finally, the concept of stationary solutions is introduced in order to motivate the following chapter.

During the 1970s to 2000s, more than 180 studies of the elastic thickness of the lithosphere were published. The results of these studies have provided a wealth of new information on the long-term mechanical properties of the lithosphere and their relationship to plate and load age. Although the results of individual studies are subject to uncertainties, the analysis of large, global data sets tends to ‘smooth’ out local discrepancies and, hence, make it more likely they will reveal the main features that describe the long-term behaviour of the lithosphere.

Oceanic and continental flexure studies suggest that the long-term behaviour of the lithosphere can be modelled, to first order, as a thin elastic plate that overlies an inviscid fluid. The thickness of the elastic plate, Te, varies both spatially and temporally, and this has provided information on the relationship between load and plate age.

This chapter is a collection of facts, ideas, and techniques regarding the analysis of boundary value, initial and initial boundary value problems for partial differential equations. We begin by deriving some of the representative equations of mathematical physics, which then give rise to the classification of linear, second order, constant coefficient partial differential equations into: elliptic, parabolic, and hyperbolic equations. For each one of these classes we then discuss the main ideas behind problem with them and the existence of solutions: both classical and weak.

We learn about unbound states and find that the energies are no longer quantized. We learn about momentum eigenstates and superposing momentum eigenstates in a wave packet. We apply unbound states to the problem of scattering from potential wells and barriers in one dimension.

A brief introduction to Fourier series and transforms, with explicit expressions for a selected number of functions.

This chapter discusses the transition between Fourier series and Fourier Transform, which is the tool for spectrum analysis. Generally, the use of linearly independent base functions allows a wide range of linear regression models that work in a least square sense such that the total error squared is minimized in finding the coefficients of the base functions. A special case is sinusoidal functions based on a fundamental frequency and all its harmonics up to infinity. This leads to the Fourier series for periodic functions. In this chapter, we start from the original Fourier series expression and convert the sinusoidal base functions to exponential functions. We can then consider the limit when the length of the function and the period of the original function approach infinity (so that the fundamental frequency approaches 0, including aperiodic functions), leading to the Fourier integral and Fourier Transform. We can then define the inverse Fourier Transform and establish the relationship between the coefficients of Fourier series and the discrete form Fourier Transform. All these are preparations for the fast Fourier Transform (FFT), an efficient algorithm of computation of the discrete Fourier Transform that is widely used in data analysis for oceanography and other applications.

Based on our preceding discussions of atomic-resolution characterization techniques in Chapter 4, no technique has yet achieved ASAT. Combining information from FIM or (S)TEM along with APT has demonstrated some very promising results, and each combination seems to be a likely path toward ASAT. In this chapter, we propose how ASAT might be achieved using correlative and/or combined techniques such as (S)TEM + APT. Such a combination would allow several routes for determination of the ion transfer function, or how imaging occurs during an APT experiment. If we can determine the transfer function with high-enough fidelity, we make the argument that it should be possible to achieve ASAT using a combination of (S)TEM and APT with inputs from simulations.

We give a short introduction to some rigorous aspects of distribution theory.

This chapter provides an introduction to the notion of physical dimension, to the specific notations which are used in physics, as well a brief review of some basic mathematics: an introduction to informal distribution theory, to the delta function and the Fourier transform.