II-1
dx
i
±
±
f
±
q
1
dq
1
²
±
±
2
2
²
±
±
3
3
±
²
j
±
x
±
also
±
±
±
²
k
±
±
±
±
using chain rule
±
±
±
²
A
.
±
±
±
±
±
±
±
²
B
.
±
±
Chapter II: General Coordinate Transformations
Consider two coordinate systems in 3-dimensional Euclidian space:
1. a Cartesian system where a point is specified by (x
1
, x
2
, x
3
)
±
(x,y,z)
2. a general "q
i
" system where a point is specified by
(q
1
, q
2
, q
3
).
Each point (x
1
, x
2
, x
3
) corresponds to a unique set of real numbers (q
1
, q
2
, q
3
).
Further, each x
i
is a function of the q
j
,
f
i
(q
1
,q
2
,q
3
), and each q
j
= h
j
(x
1
,x
2
,x
3
) where all first partial derivatives of f
i
and h
j
exist.
Using only the chain rule
for differentiation, the following equations can be obtained:
where we have used the notation x
i
(q
1
,q
2
,q
3
)
±
f
i
(q
1
,q
2
,q
3
) and q
j
(x
1
,x
2
,x
3
)
±
h
j
(x
1
,x
2
,x
3
) .
Note that the differential, dx
i
, "transforms with" [
²
x
i
/
²
q
j
]
and the partial derivative "transforms with"
[
²
q
j
/
²
x
i
].
So, using the summation notation (repeated indices ==> summation):
Note the placement of the indices.