The previous chapter contains several ideas that are important to a full understanding of tensors. The first is that any vector may be represented by components that transform between coordinate systems in one of two ways. “Covariant” components transform in the same manner as the original basis vectors pointing along the coordinate axes, and “contravariant” components transform in the inverse manner of those basis vectors. The second main idea is that coordinate basis vectors are tangent to the coordinate axes, and that there also exist reciprocal or dual basis vectors that are perpendicular to the coordinate axes; these dual basis vectors transform inversely to the coordinate basis vectors. The third idea is that combining contravariant components with original basis vectors and combining covariant components with dual basis vectors produces a result that is invariant under coordinate transformation. That result is the vector itself, and the vector is the same no matter which coordinate system you use for its components.

This chapter extends the concepts of covariance and contravariance beyond vectors and makes it clear that scalars and vectors are members of the class of objects called “tensors.”

Definitions (advanced)

In the basic definitions of Chapter 1, scalars, vectors, and tensors were defined by the number of directions involved: zero for scalars, one for vectors, and more than one for tensors.

This chapter provides examples of how to apply the tensor concepts contained in Chapters 4 and 5, just as Chapter 3 provided examples of how to apply the vector concepts presented in Chapters 1 and 2. As in Chapter 3, the intent for this chapter is to include more detail about a small number of selected applications than can be included in the chapters in which tensor concepts are first presented.

The examples in this chapter come from the fields of Mechanics, Electromagnetics, and General Relativity. Of course, there's no way to comprehensively cover any significant portion of those fields in one chapter; these examples were chosen only to serve as representatives of the types of tensor application you're likely to encounter in those fields.

The inertia tensor

A very useful way to think of mass is this: mass is the characteristic of matter that resists acceleration. This means that it takes a force to change the velocity of any object with mass. You may find it helpful to think of moment of inertia as the rotational analog of mass. That is, moment of inertia is the characteristic of matter that resists angular acceleration, so it takes a torque to change the angular velocity of an object.

Many students find that rotational motion is easier to understand by keeping the relationships between translational and rotational quantities in mind.

The vector concepts and techniques described in the previous chapters are important for two reasons: they allow you to solve a wide range of problems in physics and engineering, and they provide a foundation on which you can build an understanding of tensors (the “facts of the universe”). To achieve that understanding, you'll have to move beyond the simple definition of vectors as objects with magnitude and direction. Instead, you'll have to think of vectors as objects with components that transform between coordinate systems in specific and predictable ways. It's also important for you to realize that vectors can have more than one kind of component, and that those different types of component are defined by their behavior under coordinate transformations.

So this chapter is largely about the different types of vector component, and those components will be a lot easier to understand if you have a solid foundation in the mathematics of coordinate-system transformation.

Coordinate-system transformations

In taking the step from vectors to tensors, a good place to begin is to consider this question: “What happens to a vector when you change the coordinate system in which you're representing that vector?” The short answer is that nothing at all happens to the vector itself, but the vector's components may be different in the new coordinate system. The purpose of this section is to help you understand how those components change.

The real value of understanding vectors and how to manipulate them becomes clear when you realize that your knowledge allows you to solve a variety of problems that would be much more difficult without vectors. In this chapter, you'll find detailed explanations of four such problems: a mass sliding down an inclined plane, an object moving along a curved path, a charged particle in an electric field, and a charged particle in a magnetic field. To solve these problems, you'll need many of the vector concepts and operations described in Chapters 1 and 2.

Mass on an inclined plane

Consider the delivery woman pushing a heavy box up the ramp to her delivery truck, as illustrated in Figure 3.1. In this situation, there are a number of forces acting on the box, so if you want to determine how the box will move, you need to know how to work with vectors. Specifically, to solve problems such as this, you can use vector addition to find the total force acting on the box, and then you can use Newton's Second Law to relate that total force to the acceleration of the box.

To understand how this works, imagine that the delivery woman slips off the side of the ramp, leaving the box free to slide down the ramp under the influence of gravity.

Computer simulation is an indispensible research tool for modeling, understanding, and predicting nanoscale phenomena. There is a huge gap between the complexity of the programs and algorithms used in computational physics courses and and those used in research for computer simulations of nanoscale systems. The advanced computer codes used by researchers are often too complicated for students who want to develop their own codes, want to understand the essential details of computer simulations, or want to improve existing programs.

The aim of this book is to provide a comprehensive program library and description of advanced algorithms to help students and researchers learn novel methods and develop their own approaches. An important contribution of this book is that it is accompanied by an algorithm library in Fortran 90 that implements the computational approaches described in the text.

The physical problems are solved at various levels of sophistication using methods based on classical molecular dynamics, tight binding, density functional approaches, or fully correlated wave functions. Various basis functions including finite differences, Lagrange functions, plane waves, and Gaussians are introduced to solve bound state and scattering problems and to describe electronic structure and transport properties of materials. Different methods of solving the same problem are introduced and compared.

The book is divided into two parts. In the first part we concentrate on one-dimensional problems.