This book has one purpose: to help you understand vectors and tensors so that you can use them to solve problems. If you're like most students, you first encountered vectors when you took a course dealing with mechanics in high school or college. At that level, you almost certainly learned that vectors are mathematical representations of quantities that have both magnitude and direction, such as velocity and force. You may also have learned how to add vectors graphically and by using their components in the x-, y- and z-directions.

That's a fine place to start, but it turns out that such treatments only scratch the surface of the power of vectors. You can harness that power and make it work for you if you're willing to delve a bit deeper – to see vectors not just as objects with magnitude and direction, but rather as objects that behave in very predictable ways when viewed from different reference frames. That's because vectors are a subset of a larger class of objects called “tensors,” which most students encounter much later in their academic careers, and which have been called “the facts of the Universe.” It is no exaggeration to say that our understanding of the fundamental structure of the universe was changed forever when Albert Einstein succeeded in expressing his theory of gravity in terms of tensors.

If you were tracking the main ideas of Chapter 1, you should realize that vectors are representations of physical quantities – they're mathematical tools that help you visualize and describe a physical situation. In this chapter, you can read about a variety of ways to use those tools to solve problems. You've already seen how to add vectors and how to multiply vectors by a scalar (and why such operations are useful); this chapter contains many other “vector operations” through which you can combine and manipulate vectors. Some of these operations are simple and some are more complex, but each will prove useful in solving problems in physics and engineering. The first section of this chapter explains the simplest form of vector multiplication: the scalar product.

Scalar product

Why is it worth your time to understand the form of vector multiplication called the scalar or “dot” product? For one thing, forming the dot product between two vectors is very useful when you're trying to find the projection of one vector onto another. And why might you want to do that? Well, you may be interested in knowing how much work is done by a force acting on an object. The first instinct of many students is to think of work as “force times distance” (which is a reasonable starting point).

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.