eld is taken as u(r), so that
r = r + u(r) .
In its neighbourhood Q (at r + r) is displaced to Q (at r + r ). So an element
P Q r (needle) for the unstrained body moves to P Q r under the strain.
We are interested in the change in r which can be derived f
In S , the velocity of light would be direction dependent with |r | = c
Could then dene the aether as the frame where we have spherical waves with
r = cer
0 0 = 1/c2
Alternatively consider Maxwells equations directly
E =
0
B
= 0
t
B = 0
E+
B = 0 j + 0
With a body force b per unit volume (eg gravity where b = g), then if f is the total
force per unit volume acting on V , we have (dFi Fi dS = Pij nj dS Pij dSj )
fi dV
=
V
bi dV +
dFi
V
S
=
Pij
xj
bi +
V
dV ,
upon using the (generalised) divergence theore
So as P is symmetric
W dV
=
Pij
V
V
=
1
2
ui uj
+
xj
xi
dV
Pij Eij dV ,
V
as
u
u
u =
(u(r + r) u(r) =
(r + r)
(r) =
xj
xj
xj
xj
u
xj
,
and so E is a small change in strain tensor. Thus
W = Pij Eij
= cijkl Eij Ekl
= 1 cijkl (Eij Ekl ) ,
2
upon using Hook
giving
(r) r2 ij ri rj dV
Iij (O) =
(r ) (R + r )2 ij (Ri + ri )(Rj + rj ) dV
=
where we have set (r ) (r) and changed integration variables to r . Upon
expanding the integrand and using the denition of G, namely (r )r dV = 0
gives
Iij (O) =
(r ) (R2
or
i=
V
R
with R =
l
.
A
R is the resistance, measured in V /A or Ohms, .
We can generalise Ohms Law by writing
ji = ij Ej ,
where is the conductivity tensor (by the quotient theorem).
Layered material of conductor and insulator where the current can onl
is a pseudotensor of rank 3. So is isotropic or invariant (more later). Proof
def
ijk = ijk = det L det Lijk = det L li lj lk ,
(using the denition of the determinant). So is a third rank pseudotensor.
With one can build pseudotensors of higher rank:
eg
4.1.1
Some examples
eg
f (x) = sin x ,
and we wish to expand about 0. Now
f (2n) (0) = (1)n sin 0 = 0
f (2n+1) (0) = (1)n cos 0 = (1)n ,
and as |f (m) ()| 1 then
|Rm | =
1 m
1 m (m)
|x |f ()|
x 0,
m!
m!
for xed x. So
(1)n
sin x =
n=0
x2n+1
x3 x5
=x
+
+ .
Yet another form is
ijk lmn det A =
ail aim ain
ajl ajm ajn
akl akm akn
.
(Take original denition of det A as | | and permute rows/columns, this produces
signs equivalent to permutations.)
1.3.3
Linear Equations
Proto-type use of matrices/determinants. I
If Tijks is a tensor of rank n, then Tiiks (ie n 2 free indices) is a tensor of
rank n 2 the process is called contraction.
For example Tij = ai bj is a tensor of rank 2 and Tii = ai bi is a tensor of rank
0 (scalar).
This process of multiplying two tens
1.3
Matrices and Determinants
1.3.1
Matrices
An M N matrix is a rectangular array of numbers M rows and N columns,
a11
a12
a1,N 1
a1N
a21
a22
a2,N 1
a2N
= (aij ) .
A=
aM 1,1 aM 1,2
aM 1,N 1 aM 1,N
aM,1
aM,2
aM,N 1
aM,N
aij Aij with 1 i M , 1 j N are
L = rotation matrix (or matrix of direction cosines)
lij = cos of angle between ith axis of S and jth axis of S
This is the fundamental result for the rotation of a vector. Note that by this we
mean that the vector is not rotated it stays xed in space b
Chapter 1
Vectors, Matrices and
Determinants
1.1
Cartesian Vectors, and symbols
We have scalars denoted by one number, eg temperature, and vectors, characterised
by a direction and length, eg velocity. We shall consider two classes of vectors:
Displaceme
The Ti are called the generators of innitesimal rotations about basis vectors ei (the
i is conventional so Ti are Hermitian, Ti TiT = Ti ).
So innitesimal rotation through angle about n is
R(, n) = 1 i n T .
def
If [A, B] = AB BA is the commutator, then