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This textbook presents all the mathematical and physical concepts needed to visualize and understand representation surfaces, providing readers with a reliable and intuitive understanding of the behavior and properties of anisotropic materials, and a sound grasp of the directionality of material properties. They will learn how to extract quantitative information from representation surfaces, which encode tremendous amounts of information in a very concise way, making them especially useful in understanding higher order tensorial material properties (piezoelectric moduli, elastic compliance and rigidity, etc.) and in the design of applications based on these materials. Readers will also learn from scratch concepts on crystallography, symmetry and Cartesian tensors, which are essential for understanding anisotropic materials, their design and application. The book describes how to apply representation surfaces to a diverse range of material properties, making it a valuable resource for material scientists, mechanical engineers, and solid state physicists, as well as advanced undergraduates in Materials Science, Solid State Physics, Electronics, Optics, Mechanical Engineering, Composites and Polymer Science. Moreover, the book includes a wealth of worked-out examples, problems and exercises to help further understanding.

### Chapter 1. Introduction

Abstract
A representation surface (RS from now on) is a simple graphical tool that helps understand how a physical magnitude depends on direction. Without worrying for the time being about the meaning of the two objects shown in the figure below, a RS conveys the information contained in the diagram on the left by means of the shaded surface on the right hand side. Most people find it easy to determine what the symmetries of the RS are, or in which directions it is elongated or shortened.
Manuel Laso, Nieves Jimeno

### Chapter 2. Geometric Symmetry

Abstract
This is a mostly visual and almost equation-free chapter which is usually appealing to readers with good spatial intuition. Its first half is a minimal but self-contained presentation of crystallographic concepts needed in the rest of the book. The second half is devoted to the determination of the crystallographic or limit class (alias for point group) to which a material belongs, based on its geometric elements of symmetry.
Manuel Laso, Nieves Jimeno

### Chapter 3. Representation of Material Properties by Means of Cartesian Tensors

Abstract
The goal of this chapter is to familiarize the reader with the basic ideas and the few manipulation rules for Cartesian tensors [16] that will be required in the rest of the book. If you have previously been exposed to and perhaps intimidated by expressions like “covariant” and “contravariant”, you are about to make a quantum leap in your study of tensor material properties: when working with Cartesian tensors the concepts of “covariant” and “contravariant” are not necessary. The manipulation rules for Cartesian tensors are but a minor extension of the familiar rules for vector and matrices. In the rest of the book the adjective “Cartesian” will often be dropped for brevity.
Manuel Laso, Nieves Jimeno

### Chapter 4. Representation Surfaces

Abstract
A representation surface is a graphical way of visualizing the dependence of a given tensorial magnitude (both material and field tensors) on the direction in which it is measured. In loose terms, the RS is the polar spherical representation of the projection of a tensor in one (or more) directions.
Manuel Laso, Nieves Jimeno

### Chapter 5. Scalar and Rank One Properties

Abstract
In this chapter we  meet the simplest RS. But before doing so, a brief review of some basic material and some general features of RSs are presented, as well as those specific of odd and even order properties which will be useful in later chapters.
Manuel Laso, Nieves Jimeno

### Chapter 6. Symmetric Second Rank Properties

Abstract
In the previous chapter RSs were first introduced for the polar classes in their conventional axes, and then in an axes-free setting. We saw that a single type of geometrical object, the oriented sphere, was enough to construct the RS for all crystallographic and limit classes.
Manuel Laso, Nieves Jimeno

### Chapter 7. Third Rank Properties

Abstract
Odd rank properties are, in Newnham’s words, null properties, meaning that they may vanish for certain point groups (like all centrosymmetric ones, see Sect. 3.​6). As a result, not all materials will display third rank properties. Also as a consequence of being of odd rank, the RS will consist of overlapping positive and negative lobes, as shown in Fig. 7.1.
Manuel Laso, Nieves Jimeno

### Chapter 8. Fourth Rank Properties

Abstract
The extension to fourth rank tensors of the ideas presented in the previous chapter is straightforward. Properties like magnetoresistivity $$\underline{\underline{\underline{\underline{\rho }}}}^{mag}$$, the Kerr effect coefficient $$\underline{\underline{\underline{\underline{K}}}}$$, the piezo-optic coefficient $$\underline{\underline{\underline{\underline{\pi }}}}^{opt}$$, the electrostriction coefficient $$\underline{\underline{\underline{\underline{M}}}}$$, and the elastic compliance $$\underline{\underline{\underline{\underline{s}}}}$$ and stiffness $$\underline{\underline{\underline{\underline{c}}}}$$ are fourth rank properties whose RSs are defined in the same way.
Manuel Laso, Nieves Jimeno

### Chapter 9. Transversality and Skew Symmetric Properties

Abstract
The reader may have wondered why almost only RS corresponding to projections such as $$:\underline{n}\,\underline{n}\,$$, , have appeared up to now. As a matter of fact, we have also encountered projections along more than one direction: the calculation of the angular dependence of the shear modulus G (8.​22) of a 2D fabric involved the use of two vectors (8.​18). The calculation of polycrystalline averages also involved terms like and (8.​59). All these expressions come from the definition of engineering moduli (Sect. 4.​7) or similar magnitudes that involve more than one direction in their projection.
Manuel Laso, Nieves Jimeno