Mathematical Tools for Physics

12—Tensors 367

But,
~u. 11 T(~v) =uieˆi.vj^11 T(~ei)
=uivjˆei.(Tkjˆek)
=uivjTij.

The last step comes from the orthonormality of theˆe’s. Becauseuiandvjare arbitrary, this shows that

Tij=^02 T(ˆei,ˆej) (23)

This is the equation that makes the transformation of bases simple. If you want to computeTij′ this is

Tij′ =^02 T(ˆe′i,ˆe′j)

Each of theˆe′iis expressible In terms of theˆei. For example

T 11 ′ =^02 T(ˆe′ 1 ,ˆe′ 1 ) =^02 T

(

eˆ 1 +ˆe 2

√

2

,

ˆe 1 +eˆ 2

√

2

)

Use linearity in each of the variables, and you get

T 11 ′ =

1

2

[ 0

2 T(ˆe^1 ,ˆe^1 ) +

0

2 T(ˆe^1 ,ˆe^2 ) +

0

2 T(ˆe^2 ,ˆe^1 ) +

0

2 T(eˆ^2 ,ˆe^2 )

]

=

1

2

[

T 11 +T 12 +T 21 +T 22

]

This is the same result as in Eq. ( 21 ), and it equals 3/2. Another case would beT 12 ′

T 12 ′ =^02 T(ˆe′ 1 ,ˆe′ 2 ) =^02 T

(

ˆe 1 +ˆe 2

√

2

,

eˆ 2 −ˆe 1

√

2

)

=

1

2

[

T 12 −T 11 +T 22 −T 21

]

,

with the same results as before.