Question

Let I=∫∫D(x2−y2)dxdy, where

D={(x,y):3≤xy≤6,0≤x−y≤7,x≥0,y≥0}
Show that the mapping u=xy, v=x−y maps D to the rectangle R=[3,6]×[0,7].

Compute ∂(x,y)/∂(u,v) by first computing ∂(u,v)/∂(x,y).
∂(x,y)/∂(u,v)=

Answer 1

The mapping u = xy, v = x - y maps D to the rectangle R = [3,6] x [0,7].

Step-by-step explanation:

To prove that the mapping u = xy, v = x - y maps D to the rectangle R = [3,6] x [0,7], we need to show that the range of u and v corresponds to the interval [3,6] for u and the interval [0,7] for v.

1. Let's consider the range of the variable u = xy: The minimum value of xy occurs when both x and y are at their minimum values, which is 3. The maximum value of xy occurs when both x and y are at their maximum values, which is 6. Therefore, the range of u is [3,6].
2. Now, let's consider the range of the variable v = x - y: The minimum value of x - y occurs when x is at its minimum value and y is at its maximum value, which is 0. The maximum value of x - y occurs when x is at its maximum value and y is at its minimum value, which is 7. Therefore, the range of v is [0,7].
3. Therefore, the mapping u = xy, v = x - y maps D to the rectangle R = [3,6] x [0,7].