Question

Solve the system
u=2x-y v= 3 x+2y​, for x and y in terms of u and v then find the value of Jacobian σ(x,y)/σ(u,v). Find the image under the transformation of the triangular region with the vertices (0,0) ,(2,4), and (2, -3) in the x,y plane. Sketch the transformed region in uv plane.The function for x in terms of u and v is x=?

Answer 1

Answer:

x = $\frac{2u+v}{7}$

y =  $\frac{2v - 3u }{7}$

Jacobian value = 1/7

Step-by-step explanation:

Given data:

u = 2x -y

v = 3x + 2y

solving the system for  x and  y in terms of u and v

x = $\frac{2u+v}{7}$

y =  $\frac{2v - 3u }{7}$

The value of the Jacobian =  1/7 ( solution attached below )

Image under the transformation of the triangular region with vertices ( 0,0) , (2,4) and (2,-3) in the x-y plane is attached below

Answer 2

In this exercise we have to use system and Jacobian knowledge to calculate the requested values, thus we find that:

 $X= \frac{2u+v}{7} \\\\Y= \frac{2v-3u}{7}\\\\J= \frac{1}{7}$

Given data as:  

• $u = 2x -y$

• $v = 3x + 2y$

Solving the system for  x:

$2u=4x-2y\\v= 3x-2y\\= 2u+v/7$

Solving the system for  v:

$u = 2(2u+v/7)-y\\y= 1/7(2v-3u)$

Find the value of the Jacobian:

$\frac{dx}{du}= \frac{d(2u+v/7)}{du}= 2/7\\\\\frac{dv}{du}= \frac{d(2u+v/7)}{du}= 2/7$

$\frac{dx}{du}= \frac{d(2v+3u/7)}{du}= -3/7\\\\\frac{dv}{du}= \frac{d(2v+3u/7)}{du}= 2/7$

Substitute the given value in equation will become:

$J= 1/7$

See more about system at brainly.com/question/7589753