which reduces (1.6) to
(1.8)
1

2g
1
0
1 + (y )2

1 y
dx.
Clearly, changing the curve on which the bead slides down will chang
the value of the above integral, which is therefore a function of y:
is a function of a function, or a functional for short. Whenever w
wish to emphasize that a functional J depends on y, we will denot
it by J[y], as in
(1.9) J[y] =
1
0
1 +
(
y
)2
1 y
dx.
At other times, however, we may prefer to emphasize that the func
tional depends on the curve y = y(x), i.e., on the graph of y, whic
we denote by Γ; in that case, we will denote the functional by J[Γ
At other times still, we may have no particular emphasis in mind, i
which case, we will write the functional as plain old J. For example
if Γ is a straight line, then
(1.10) y(x) = 1 x,
and (1.9) yields
(1.11) J =
1
0
{1 + (−1)2}2
1

x
dx = 2

2
1
0
d
dx
{x
1
2
} dx = 2

2
or approximately 2.82843; whereas if Γ is a quarter of the circle o
radius 1 with center (1, 1), then
(1.12) y(x) = 1 2x x2
and
(1.13) J =
1
0
1
(2x x2)
3
4
dx 2.62206
on using numerical methods.3
3E.g.,
the Mathematica command NIntegrate[(2x-xˆ2)ˆ(-3/4),{x,0,1}].
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