James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 15.9 Change of Variables in Multiple Integrals 1055
u − u 0 ) is
r v − t v su 0 , v 0 d i 1 h v su 0 , v 0 d j − −x
−v i 1 −y
−v j
r (u¸, √¸+Î√)
b
r (u¸, √¸)
R
a
r (u¸+Îu, √¸)
FIGURE 4
We can approximate the image region R − TsSd by a parallelogram determined by the
secant vectors
a − rsu 0 1 Du, v 0 d 2 rsu 0 , v 0 d
shown in Figure 4. But
and so
rsu 0 1 Du, v 0 d 2 rsu 0 , v 0 d
r u − lim
Du l 0
Du
rsu 0 1 Du, v 0 d 2 rsu 0 , v 0 d < Du r u
b − rsu 0 , v 0 1 Dvd 2 rsu 0 , v 0 d
Î√r √
r (u¸, √¸)
FIGURE 5
Îu r u
Similarly
rsu 0 , v 0 1 Dvd 2 rsu 0 , v 0 d < Dv r v
This means that we can approximate R by a parallelogram determined by the vectors
Du r u and Dv r v . (See Figure 5.) Therefore we can approximate the area of R by the area
of this parallelogram, which, from Section 12.4, is
6 | sDu r ud 3 sDv r v d | − | r u 3 r v | Du Dv
Computing the cross product, we obtain
r u 3 r v −
i j k
−x −y
−u −u
0
−x −y
−v −v
0
−
−x
−u
−x
−v
−y
−u
−y
−v
k −
−x
−u
−y
−u
−x
−v
−y
−v
k
The determinant that arises in this calculation is called the Jacobian of the transforma tion
and is given a special notation.
The Jacobian is named after the
German mathematician Carl Gustav
Jacob Jacobi (1804–1851). Although
the French mathematician Cauchy
first used these special determinants
involving partial derivatives, Jacobi
developed them into a method for
evaluating multiple integrals.
7 Definition The Jacobian of the transformation T given by x − tsu, vd and
y − hsu, vd is
Z
−x −x
Z
−sx, yd
−su, vd − −u −v
− −x −y
−y −y −u −v 2 −x −y
−v −u
−u −v
With this notation we can use Equation 6 to give an approximation to the area DA
of R:
8 DA < Z
−sx, yd
−su, vd
Z Du Dv
where the Jacobian is evaluated at su 0 , v 0 d.
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