The components of a second-order tensor are conveniently visualized as a two-dimensional matrix, just as the components of a vector are sometimes visualized as a one-dimensional matrix. However, it is important to recognize that an th-order tensor is not simply another name for an -dimensional matrix. A matrix is merely an ordered set of numbers. A tensor, on the other hand, is an ordered set of components that have specific transformation properties under rotation of the coordinate axes. (See Section B.3.)

Consider two vectors and that are represented as and , respectively, in tensor notation. According to Section A.6, the scalar product of these two vectors takes the form

(B.1) |

The previous expression can be written more compactly as

Here, we have made use of the

According to Section A.8, the vector product of two vectors and takes the form

(B.3) | ||

(B.4) | ||

(B.5) |

in tensor notation. The previous expression can be written more compactly as

Here,

is known as the third-order

It is helpful to define the second-order *identity tensor* (also known as the Kroenecker delta tensor),

It is easily seen that

et cetera.

The following is a particularly important tensor identity:

In order to establish the validity of the previous expression, let us consider the various cases that arise. As is easily seen, the right-hand side of Equation (B.16) takes the values

Moreover, in each product on the left-hand side of Equation (B.16), has the same value in both factors. Thus, for a non-zero contribution, none of , , , and can have the same value as (because each factor is zero if any of its indices are repeated). Because a given subscript can only take one of three values ( , , or ), the only possibilities that generate non-zero contributions are and , or and , excluding (as each factor would then have repeated indices, and so be zero). Thus, the left-hand side of Equation (B.16) reproduces Equation (B.19), as well as the conditions on the indices in Equations (B.17) and (B.18). The left-hand side also reproduces the values in Equations (B.17) and (B.18) because if and then and the product (no summation) is equal to , whereas if and then and the product (no summation) is equal to . Here, use has been made of Equation (B.8). Hence, the validity of the identity (B.16) has been established.

In order to illustrate the use of Equation (B.16), consider the vector triple product identity (see Section A.11)

In tensor notation, the left-hand side of this identity is written

(B.21) |

where use has been made of Equation (B.6). Employing Equations (B.8) and (B.16), this becomes

(B.22) |

which, with the aid of Equations (B.2) and (B.13), reduces to

(B.23) |

Thus, we have established the validity of the vector identity (B.20). Moreover, our proof is much more rigorous than that given earlier in Section A.11.