Page 43 - Intro to Tensor Calculus

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and

r
r
r
∂~ ∂~ ∂~
dw = grad w · du + gradw · dv + grad w · dw. (1.2.21)
∂u ∂v ∂w
By comparing like terms in equations (1.2.20) and (1.2.21) we find
~ 2 ~
~ 2 ~
~ 2 ~
E · E 1 =0, E · E 2 =1, E · E 3 =0
(1.2.22)
~ 3 ~
~ 3 ~
~ 3 ~
E · E 1 =0, E · E 2 =0, E · E 3 =1.
The equations (1.2.22) and (1.2.19) show us that the basis vectors defined by equations (1.2.12) and (1.2.13)
are reciprocal.
Introducing the notation
1 2 3 1 2 3
(x ,x ,x )= (u, v, w) (y ,y ,y )= (x, y, z) (1.2.23)

where the x s denote the generalized coordinates and the y s denote the rectangular Cartesian coordinates,
0
0
the above equations can be expressed in a more concise form with the index notation. For example, if
1
1
2
i
i
i
i
i
2
i
3
3
x = x (x, y, z)= x (y ,y ,y ), and y = y (u, v, w)= y (x ,x ,x ), i =1, 2, 3 (1.2.24)
then the reciprocal basis vectors can be represented
i
~ i
E = gradx , i =1, 2, 3 (1.2.25)
and
∂~ r
~
E i = , i =1, 2, 3. (1.2.26)
∂x i
3
2
1
r
r
We now show that these basis vectors are reciprocal. Observe that ~ = ~(x ,x ,x )with
r
∂~ m
r
d~ = dx (1.2.27)
∂x m
and consequently
r
∂~
i i i m ~ i ~ m i m
dx = grad x · d~r = gradx · m dx = E · E m dx = δ dx , i =1, 2, 3 (1.2.28)
m
∂x
Comparing like terms in this last equation establishes the result that
~ i ~
i
E · E m = δ , i, m =1, 2, 3 (1.2.29)
m
which demonstrates that the basis vectors are reciprocal.