Vector Analysis

Carlyle Moore

Vectors were used extensively in the treatment of the basic laws of Electricity and Magnetism, relying on the notion of a vector as a quantity that is characterized by a numerical magnitude and a direction. This notion may be supplanted by a more sophisticated definition, in terms of the transformation properties of the vector components. Indeed, vectors (and scalars too) can be shown to belong to a wider class of mathematical objects known as tensors, which are defined by their transformation properties. While this refinement will not be pursued here, the analysis can be simplified considerably by the use of the tensor notation. The student will find the saving in labor afforded by this approach to be quite dramatic. Our main objective is to re-formulate the laws of Electricity and Magnetism in differential form, a procedure which allows us to demonstrate the existence and properties of electromagnetic waves. As we shall see, this also lays the foundation for the treatment of optical phenomena.

Tensor Notation

The Cartesian representation of a vector may be expressed in a more compact form by replacing the components by and the unit vectors by , respectively. Thus we write

The notation may be made even more compact by omitting the summation sign altogether, i.e. by invoking the Einstein summation convention, which prescribes a summation over each repeated index. Using this convention, Equation (1.1) may be written in the form

The components of the position vector r are usually denoted by , so we write

Clearly, the choice of index for a given summation is immaterial, but the same index cannot be used for more than one summation in the same expression.

The Scalar Product

The notation introduced above allows us to write the scalar (dot) product of two vectors A and B as follows:

Two Useful Symbols

Vector analysis may be considerably simplified by the use of the Kronecker delta and the Levi-Civita, or permutation, symbol . The Kronecker delta is defined as follows:

It follows that , i.e. is symmetric. As an example of its use, let us consider the partial derivative . If , the derivative is clearly equal to 1. On the other hand, if , then are independent of each other, hence the derivative is equal to zero. In other words,

The Kronecker delta frequently appears in summations over the index j or k, or both. These may easily be evaluated by noting that each term in the summation is zero unless . For example, represents a summation over the (repeated) index j, the only non-zero term of which is the term for which . From the definition (1.5), it is clear that

Similarly,

The permutation symbol is totally anti-symmetric, i.e. it changes sign upon interchange of any two of the indices. It follows that is equal to zero if any two of the indices are equal, and (where n is non-zero) when all the indices are different. By convention, we choose . Note that does not change sign if the indices appear in the same cyclic order, i.e. . Now it is often required to evaluate the product of two permutation symbols with a common index. The result is

Note that the common index i appears in the same location in both terms on the left hand side.

Another type of product commonly encountered in vector analysis is of the form , where is symmetric, and is anti-symmetric, in the indices j and k. Since j and k are both summation (dummy) indices, the product may be written as follows:

i.e.

(1.12)

The Vector Product

We define the vector (cross) product of two vectors A and B as follows:

Since this definition of the vector product may be unfamiliar to the reader, we shall demonstrate that it is equivalent to the conventional definition encountered in elementary texts. To do this, we note, first, that

i.e.

where q is the angle between the vectors A and B. Thus

whence

Note also that

since is symmetric, and is anti-symmetric, in the indices i and j. Similarly, it can be shown that . In other words, is perpendicular to both A and B, i.e. it is perpendicular to the plane defined by A and B.

The sense of the vector product is given by the corkscrew rule. Choose the plane defined by the vectors A and B as the x-y plane, with A pointing in the positive x-direction and B in the upper half plane, as shown. The vector A must then be rotated anti-clockwise through an angle in order to make it point in the direction of B.

Thus, according to the corkscrew rule, the vector should point in the positive z-direction. We know from the discussion above that is perpendicular to the x-y plane, and from Equation (1.13), we have

since are both positive. This shows that points in the positive z-direction, as required by the corkscrew rule.

Example 1.1

Evaluate the following: (i) (ii) (iii)

(i) Since is anti-symmetric, and is symmetric, in the indices j and k, it follows from Equation (1.12) that

Alternatively, we may note that if , then ; whereas if , then .

(ii) Using Equation (1.9), we have

where we have used the result .

(iii) Similarly, we have

Example 1.2

Given two perpendicular vectors A and B and a scalar , find the vector P which satisfies the relations:

Note first that the vector P is not uniquely determined by , since this relation involves two independent parameters: P and the angle between A and P. In component form, we have

Multiplying both sides of Equation (i) by (and summing over the index m), we get

Using Equation (1.9), this reduces to

(iii)

(iv)

i.e.

(v)

(vi)