Begin2.DVI

Let M(u, v ) =

1

2

(

x

∂y

∂u

−y

∂x

∂u

)

and N(u, v ) =

1

2

(

x

∂y

∂v

−y

∂x

∂v

)

and apply Green’s

theorem to the integral (8.30). Using the results

∂M

∂v =

1

2

[

x∂

(^2) y
∂u∂v +
∂x
∂v
∂y
∂u −y
∂^2 x
∂u∂v −
∂y
∂v
∂x
∂u
]

and ∂N∂u =^12

[

x ∂

(^2) y
∂v∂u +
∂x
∂u
∂y
∂v −y
∂^2 x
∂v∂u −
∂y
∂u
∂x
∂v
]

one finds (

∂N

∂u −

∂M

∂v

)

=

∣∣

∣∣

∂x

∂u

∂y

∂x ∂u

∂v

∂y

∂v

∣∣

∣∣=∂x

∂u

∂y

∂v −

∂x

∂v

∂y

∂u =J

(

x, y

u, v

)

, (8 .31)

where the determinant J is called the Jacobian determinant of the transformation

from (x, y )to (u, v ).The area integral can then be expressed in the form

A=

∫∫

R

dx dy =

∫∫

u,v

J

(

x, y

u, v

)

du dv, (8 .32)

where the limits of integration are over that range of the variables u, v which define

the region R.

Example 8-8. In changing from rectangular coordinates (x, y )to polar coordi-

nates (r, θ)the transformation equations are

x=x(r, θ) = rcos θ y =y(r, θ) = rsin θ,

and the Jacobian of this transformation is

J

(

x, y

u, v

)

=

∣∣

∣∣

∂x

∂u

∂y

∂x ∂u

∂v

∂y

∂v

∣∣

∣∣=

∣∣

∣∣ cos θ sin θ

−rsin θ r cos θ

∣∣

∣∣=r

and the area can be expressed as

∫∫

Rxy

dx dy =

∫∫

Rrθ

r dr dθ (8 .33)

which is the familiar area integral from polar coordinates.

In general, an integral of the form

∫∫

Rxy

f(x, y )dx dy under a change of variables

x=x(u, v ), y =y(u, v )becomes

∫∫

Ruv

f(x(u, v ), y (u, v )) J

(

x, y

u, v

)

du dv, where the integrand

is expressed in terms of uand vand the element of area dx dy is replaced by the new

element of area J

(x,y

u,v

)

du dv.