Question

if x=2u-v2,y=u+v2,then d(x,y)/d(u,v)=
​

Answer 1

Answer:

Step-by-step explanation:16.9 Change of Variables in Multiple Integrals

Recall: For sinlge variable, we change variables x to u in an integral by the formula

(substitution rule)

Z b

a

f(x)dx =

Z d

c

f(x(u))dx

dudu

where x = x(u), dx =

dx

dudu, and the interval changes from [a, b] to [c, d] = [x

−1

(a), x−1

(b)].

Why do we do change of variables?

1. We get a simpler integrand.

2. In addition to converting the integrand into something simpler it will often also

transform the region into one that is much easier to deal with.

notation: We call the equations that define the change of variables a transformation.

Example Determine the new region that we get by applying the given transformation

to the region R.

(a) R is the ellipse x

2 +

y

2

36 = 1 and the transformation is x =

u

2

, y = 3v.

(b) R is the region bounded by y = −x + 4, y = x + 1, and y = x/3 − 4/3 and the

transformation is x =

1

2

(u + v), y =

1

2

(u − v)

Soln:

(a) Plug the transformation into the equation for the ellipse.

(

u

2

)

2 +

(3v)

2

36

= 1

u

2

4

+

9v

2

36

= 1

u

2 + v

2 = 4

After the transformation we had a disk of radius 2 in the uv-plane.

(b)

Plugging in the transformation gives:

y = −x + 4 ⇒

1

2

(u − v) = −

1

2

(u + v) ⇒ u = 4

y = x + 1 ⇒

1

2

(u − v) = 1

2

(u + v) + 1 ⇒ v = −1

y = x/3 − 4/3 ⇒

1

2

(u − v) = 1

3

1

2

(u + v) − 4/3 ⇒ v =

u

2

+ 2

See Fig. 1 and Fig. 2 for the original and the transformed region.

Note: We can not always expect to transform a specific type of region (a triangle for

example) into the same kind of region.

Pls Brainlest this took so long