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}].

(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}].