The text I'm reading has a proof of the tensor algebra quotient rule with the following logic:
Let a[sub:2v6dwthx]k[/sub:2v6dwthx] be an arbitrary vector, and X[sub:2v6dwthx]ij[/sub:2v6dwthx] a second order tensor that satisifies:
X[sub:2v6dwthx]ij[/sub:2v6dwthx]a[sub:2v6dwthx]k[/sub:2v6dwthx] = b[sub:2v6dwthx]ijk[/sub:2v6dwthx]
where b[sub:2v6dwthx]ijk[/sub:2v6dwthx] is a third-order tensor.
Then under an orthogonal change of co-ordinates:
X'[sub:2v6dwthx]ij[/sub:2v6dwthx]a'[sub:2v6dwthx]k[/sub:2v6dwthx] = b'[sub:2v6dwthx]ijk[/sub:2v6dwthx] (1)
So far so good.
It then uses the transformation laws to arrive at:
X'[sub:2v6dwthx]ij[/sub:2v6dwthx]l[sub:2v6dwthx]km[/sub:2v6dwthx]a[sub:2v6dwthx]m[/sub:2v6dwthx] = l[sub:2v6dwthx]im[/sub:2v6dwthx]l[sub:2v6dwthx]jn[/sub:2v6dwthx]l[sub:2v6dwthx]kp[/sub:2v6dwthx]b[sub:2v6dwthx]mnp[/sub:2v6dwthx] (2)
(where l[sub:2v6dwthx]ij[/sub:2v6dwthx] is an entry in the transformation matrix)
Looking at the LHS of equations (1) and (2) above, I can see that
X'[sub:2v6dwthx]ij[/sub:2v6dwthx]a'[sub:2v6dwthx]k[/sub:2v6dwthx] = X'[sub:2v6dwthx]ij[/sub:2v6dwthx]l[sub:2v6dwthx]km[/sub:2v6dwthx]a[sub:2v6dwthx]m[/sub:2v6dwthx]
(using the general transformation law for a vector v'[sub:2v6dwthx]i[/sub:2v6dwthx] = l[sub:2v6dwthx]ij[/sub:2v6dwthx]v[sub:2v6dwthx]j[/sub:2v6dwthx])
Now looking of the RHS of both equations I can see that
b'[sub:2v6dwthx]ijk[/sub:2v6dwthx] = l[sub:2v6dwthx]im[/sub:2v6dwthx]l[sub:2v6dwthx]jn[/sub:2v6dwthx]l[sub:2v6dwthx]kp[/sub:2v6dwthx]b[sub:2v6dwthx]mnp[/sub:2v6dwthx]
(general form of the tensor transformation law)
What isn't clear is why the index of a on the LHS of (2) is the same
as for b on the RHS. The choice of index names on the RHS - m,n,p -
seems arbitrary. Why was the index m of b on the RHS matched to the index of a on the
LHS, and why specifically m (and not n or p)?
I'm new to Tensor Algebra and am trying to see this as a simple algebra task, ie how to apply simple rules to indexes when dealing with such equations.
Let a[sub:2v6dwthx]k[/sub:2v6dwthx] be an arbitrary vector, and X[sub:2v6dwthx]ij[/sub:2v6dwthx] a second order tensor that satisifies:
X[sub:2v6dwthx]ij[/sub:2v6dwthx]a[sub:2v6dwthx]k[/sub:2v6dwthx] = b[sub:2v6dwthx]ijk[/sub:2v6dwthx]
where b[sub:2v6dwthx]ijk[/sub:2v6dwthx] is a third-order tensor.
Then under an orthogonal change of co-ordinates:
X'[sub:2v6dwthx]ij[/sub:2v6dwthx]a'[sub:2v6dwthx]k[/sub:2v6dwthx] = b'[sub:2v6dwthx]ijk[/sub:2v6dwthx] (1)
So far so good.
It then uses the transformation laws to arrive at:
X'[sub:2v6dwthx]ij[/sub:2v6dwthx]l[sub:2v6dwthx]km[/sub:2v6dwthx]a[sub:2v6dwthx]m[/sub:2v6dwthx] = l[sub:2v6dwthx]im[/sub:2v6dwthx]l[sub:2v6dwthx]jn[/sub:2v6dwthx]l[sub:2v6dwthx]kp[/sub:2v6dwthx]b[sub:2v6dwthx]mnp[/sub:2v6dwthx] (2)
(where l[sub:2v6dwthx]ij[/sub:2v6dwthx] is an entry in the transformation matrix)
Looking at the LHS of equations (1) and (2) above, I can see that
X'[sub:2v6dwthx]ij[/sub:2v6dwthx]a'[sub:2v6dwthx]k[/sub:2v6dwthx] = X'[sub:2v6dwthx]ij[/sub:2v6dwthx]l[sub:2v6dwthx]km[/sub:2v6dwthx]a[sub:2v6dwthx]m[/sub:2v6dwthx]
(using the general transformation law for a vector v'[sub:2v6dwthx]i[/sub:2v6dwthx] = l[sub:2v6dwthx]ij[/sub:2v6dwthx]v[sub:2v6dwthx]j[/sub:2v6dwthx])
Now looking of the RHS of both equations I can see that
b'[sub:2v6dwthx]ijk[/sub:2v6dwthx] = l[sub:2v6dwthx]im[/sub:2v6dwthx]l[sub:2v6dwthx]jn[/sub:2v6dwthx]l[sub:2v6dwthx]kp[/sub:2v6dwthx]b[sub:2v6dwthx]mnp[/sub:2v6dwthx]
(general form of the tensor transformation law)
What isn't clear is why the index of a on the LHS of (2) is the same
as for b on the RHS. The choice of index names on the RHS - m,n,p -
seems arbitrary. Why was the index m of b on the RHS matched to the index of a on the
LHS, and why specifically m (and not n or p)?
I'm new to Tensor Algebra and am trying to see this as a simple algebra task, ie how to apply simple rules to indexes when dealing with such equations.