Starting from the definition of tensorial objects by their response to coordinate transformation, this chapter builds the flat space vector calculus machinery needed to understand the role of the metric and its associated geodesic curves in general. The emphasis here is on using tensors to build equations that are “generally covariant,” meaning that their content is independent of the coordinate system used to express them. Motivated by the transformation of gravitational energy sources, the gravitational field should be a second-rank tensor, and given the way in which that tensor must show up in a particle motion Lagrangian, it is natural to interpret that tensor as a metric.

The mathematics required to analyse higher dimensional curved spaces and space-times is developed in this chapter. General coordinate transformations, tangent spaces, vectors and tensors are described. Lie derivatives and covariant derivatives are motivated and defined. The concepts of parallel transport and a connection is introduced and the relation between the Levi-Civita connection and geodesics is elucidated. Christoffel symbols the Riemann tensor are defined as well as the Ricci tensor, the Ricci scalar and the Einstein tensor, and their algebraic and differential properties are described (though technical details of the derivationa of the Rimeann tensor are let to an appendix).

If we accept that transport barriers should be material features for experimental verifiability, we must also remember a fundamental axiom of mechanics: material response of any moving continuum, including fluids, must be frame-indifferent.This means that the conclusions of different observers regarding material behavior must transform into each other by exactly the same rigid-body transformation that transforms the frames of the observers into each other. This requirement of the frame-indifference of material response is called objectivity in classical continuum mechanics. Its significance in fluid mechanics is often overlooked or forgotten, which prompts us to devote a whole chapter to this important physical axiom. We clarify some common misunderstandings of the principle of objectivity in fluid mechanics and discuss in detail the mathematical requirements imposed by objectivity on scalars, vectors and tensors to be used in describing transport barriers.

The application of gravity gradient measurements to exploration has been growing over the past 20 years. The ability of tensor gradiometry instruments to greatly improve signal/noise when deployed on mobile platforms has transformed the usefulness of this technology. Airborne and marine Full Tensor Gradiometry (FTG) surveys have become an increasingly common part of the exploration and production toolkit. The ability of the modern instruments to provide high-resolution, spatial accuracy and very good signal/noise data has made this technology a more common part of integrated exploration and production management. The technology has a distinct cost advantage over seismic data acquisition and as such can deliver a competitive solution for imaging problems in some circumstances. There are now numerous published examples of effective use of FTG in the oil industry. The development of better instruments such as integration of direct contemporaneous measurement of conventional gravity is encouraging more interest in the technology. The potential for extending the use of FTG to reservoir monitoring and carbon dioxide sequestration assurance is likely to increase the popularity of the technology in future.

We cannot miss deep learning in a modern pattern recognition textbook, and we introduce CNN (convolutional neural networks) in this chapter. Although the mathematical derivation of CNN, especially the back-propagation process and gradient computation, is complex, we use a lot of useful tools to help readers understand what exactlyis going on in a CNN. Hence, this chapter focuses on accessibility rather than completeness. In its exercise problems, we introduce more relevant topics and methods.