Answer:
The answer is 7.25. That is there will be no change.
Step-by-step explanation:
The equations of differential vector transformation from the v-u plane to the x-y plane is given by,
[tex]\left[\begin{array}{ccc}dx\\dy\end{array}\right] = \left[\begin{array}{ccc}dx/dv&dx/du\\dy/dv&dy/du\end{array}\right] \left[\begin{array}{ccc}dv\\du\end{array}\right][/tex]
where the square matrix given is the Jacobian matrix and we can evaluate it by differentiating the equation for x and y given in the question by v and u and we will obtain the following matrix,
[tex]J = \left[\begin{array}{ccc}-4&-3\\-5&-4\end{array}\right][/tex] .
Differential area transformation from v-u coordinate system to the x-y coordinate systen is given by,
[tex]dxdy = Det(J)dvdu[/tex]
where Det(J), is evaluated as below,
[tex]Det(J) = (-4)*(-4) - (-3)*(-5) = 1[/tex]
which we plug into the above equation to obtain,
[tex]dxdy = dvdu[/tex]
in part b) area of the region D in the u-v plane is given as 7.25 that is dvdu=7.25 , that means according to the above equation that the area of the corresponding transformed region in the x-y plane that is dxdy will also be the same according o the above equation,So,
[tex]Area T(D) = dxdy = dvdu =Area D= 7.25[/tex]
a. The magnitude of the Jacobian is |J(x,y)/(u,v)| = |- 1| = 1
b. The area in region T(D∗) in the xy-plane is 7.25
Since x = -4v - 3u - 1 and y = 3 - 4u - 5v
The magnitude of the Jacobian is |J(x,y)/(u,v)| = |- 1| = 1
The Jacobian J(x,y)/(u,v) = dx/du × dy/dv - dx/dv × dy/du
Since x = -4v - 3u - 1
dx/du = d(-4v - 3u - 1)/du = -3, dx/dv = d(-4v - 3u - 1)/dv = -4
Also, y = 3 - 4u - 5v
dy/du = d(3 - 4u - 5v)/du = - 4 and dy/dv = d(3 - 4u - 5v)/dv = - 5
Substituting these values into the Jacobian, we have
J(x,y)/(u,v) = dx/du × dy/dv - dx/dv × dy/du
J(x,y)/(u,v) = - 3 × - 5 - (-4) × (-4)
J(x,y)/(u,v) = 15 - 16
J(x,y)/(u,v) = - 1
So, the magnitude of the Jacobian is |J(x,y)/(u,v)| = |- 1| = 1
The area in region T(D∗) in the xy-plane is 7.25
Since the area in the region D∗ in the uv-plane is A = 7.25,
The area in region T(D∗) in the xy-plane is A' = |J(x,y)/(u,v)|A
A' = 1 × A
A' = A
A' = 7.25
The area in region T(D∗) in the xy-plane is 7.25
Learn more about Jacobian here:
https://brainly.com/question/19427145
