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12—Tensors 299

Look atT(~v)more closely in terms of the components

T(~v) =T(v 1 eˆ 1 +v 2 eˆ 2 +v 3 ˆe 3 )

=v 1 T(ˆe 1 ) +v 2 T(ˆe 2 ) +v 3 T(eˆ 3 )

(by linearity). Each of the three objectsT(ˆe 1 ), T(ˆe 2 ), T

(

ˆe 3

)

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

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

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

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

or more compactly, T(ˆei) =

∑

j

Tjieˆj (12.13)

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 indices has been chosen for later convenience, with the sum on the first index of

theTji. This equation isthe fundamental equationfrom which everything else is derived. (It will be

modified when non-orthonormal bases are introduced later.)

Now, take these expressions forT(eˆi)and plug them back into the equation~u=T(~v):

u 1 ˆe 1 +u 2 ˆe 2 +u 3 ˆe 3 =T(~v) = v 1

[

T 11 eˆ 1 +T 21 ˆe 2 +T 31 eˆ 3

]

+v 2

[

T 12 eˆ 1 +T 22 ˆe 2 +T 32 eˆ 3

]

+v 3

[

T 13 eˆ 1 +T 23 ˆe 2 +T 33 eˆ 3

]

=

[

T 11 v 1 +T 12 v 2 +T 13 v 3

]

ˆe 1

+

[

T 21 v 1 +T 22 v 2 +T 23 v 3

]

ˆe 2

+

[

T 31 v 1 +T 32 v 2 +T 33 v 3

]

ˆe 3

Comparing the coefficients of the unit vectors, you get the relations among the components

u 1 =T 11 v 1 +T 12 v 2 +T 13 v 3

u 2 =T 21 v 1 +T 22 v 2 +T 23 v 3

u 3 =T 31 v 1 +T 32 v 2 +T 33 v 3

(12.14)

More compactly:

ui=

∑^3

j=1

Tijvj or





u 1

u 2

u 3



=





T 11 T 12 T 13

T 21 T 22 T 23

T 31 T 32 T 33









v 1

v 2

v 3



 (12.15)

At this point it is convenient to use the summation convention (first* version). This convention
says that if a given term contains a repeated index, then a summation over all the possible values of
that index is understood. With this convention, the previous equation is

ui=Tijvj. (12.16)

Notice how the previous choice of indices has led to the conventional result, with the first index denoting
the row and the second the column of a matrix.

* See section12.5for the later modification and generalization.