Sketch D and g(D) from the description of D and change of variables (x, y) = g(u, v).
-$\mathrm{x}=\mathrm{u}^{2}-\mathrm{v}^{2}, \mathrm{y}=2 \mathrm{uv} \text { where } \mathrm{D} \text { is the rectangle } 0 \leq \mathrm{u} \leq 1,0 \leq \mathrm{v} \leq 1$

B) None of these.

Question

Sketch D and g(D) from the description of D and change of variables (x, y) = g(u, v).
-$\mathrm{x}=\mathrm{u}^{2}-\mathrm{v}^{2}, \mathrm{y}=2 \mathrm{uv} \text { where } \mathrm{D} \text { is the rectangle } 0 \leq \mathrm{u} \leq 1,0 \leq \mathrm{v} \leq 1$

B) None of these.

Quizplus · Accepted Answer

From the given description of D and change of variables, we can see that x and y can be expressed in terms of u and v as:
x = u^2 - v^2;      y = 2uv
Solving these equations for u and v, we get:
u = sqrt((x + sqrt(x^2 + y^2))/2)
v = sign(y) * sqrt((-x + sqrt(x^2 + y^2))/2)
Using these equations we can sketch the transformed region g(D) in the uv-plane, which turns out to be a unit square with vertices at (0,0), (1,0), (0,1), and (1,1). The image of the rectangle with vertices (0,0), (1,0), (1,1), and (0,1) is the region bounded by the curves u = v, u = 1, and v = 1-u, which is the shaded triangle shown in the xy-plane. Therefore, the correct sketch is D.

Sketch D and G(D) from the Description of D and Change