Mathematical Tools for Physics

12—Tensors 362

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 for^11 T(ˆei)and plug them back into the equation~u=^11 T(~v):

u 1 ˆe 1 +u 2 ˆe 2 +u 3 ˆe 3 =^11 T(~v) = v 1

[

T 11 ˆe 1 +T 21 ˆe 2 +T 31 ˆe 3

]

+v 2

[

T 12 ˆe 1 +T 22 ˆe 2 +T 32 ˆe 3

]

+v 3

[

T 13 ˆe 1 +T 23 ˆe 2 +T 33 ˆe 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.

(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



. (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. (16)

* See section12.4for the later modification and generalization.