Mathematical Tools for Physics

12—Tensors 361

computation in an arbitrary basis, but for the moment it’s a little simpler to start out with the more common
orthonormal basis vectors. (Recall that an orthonormal basis is an independent set of orthogonal unit vectors,
such asˆx,ˆy,ˆz.) Some of this material was developed in chapter seven, but I’ll duplicate some of it. Start off by
examining a second rank tensor, viewed as a vector valued function

~u=^11 T(~v).

The vector~vcan be written in terms of the three basis vectors linexˆ,yˆ,zˆ. Or, as I shall denote themˆe 1 ,ˆe 2 ,eˆ 3
where
|ˆe 1 |=|ˆe 2 |=|ˆe 3 |= 1, and ˆe 1 .ˆe 2 = 0 etc. (10)

In terms of these independent vectors,~vhas componentsv 1 ,v 2 ,v 3 :

~v=v 1 ˆe 1 +v 2 ˆe 2 +v 3 ˆe 3. (11)

The vector~u=^11 T(~v)can also be expanded in the same way:

~u=u 1 ˆe 1 +u 2 ˆe 2 +u 3 ˆe 3. (12)

Look at^11 T(~v)more closely in terms of the components

1

1 T(~v) =

1

1 T(v^1 ˆe^1 +v^2 ˆe^2 +v^3 ˆe^3 )

=v 111 T(ˆe 1 ) +v 211 T(ˆe 2 ) +v 311 T(ˆe 3 )

(by linearity). Each of the three objects^11 T(ˆe 1 ),^11 T(ˆe 2 ),^11 T

(

ˆe 3

)

is a vector, which means that you can expand
each one in terms of the original unit vectors

1

1 T(ˆe^1 ) =T^11 ˆe^1 +T^21 eˆ^2 +T^31 ˆe^3

1

1 T(ˆe^2 ) =T^12 ˆe^1 +T^22 eˆ^2 +T^32 ˆe^3

1

1 T(ˆe^3 ) =T^13 ˆe^1 +T^23 eˆ^2 +T^33 ˆe^3

or more compactly,^11 T(ˆei) =

∑

j

Tjiˆej. (13)

The numbersTij(i,j= 1, 2, 3) are called the components of the tensor in the given basis. These numbers will
depend on the basis chosen, just as do the numbersvi, the components of the vector~v. The ordering of the