Question

Suppose a change of coordinates T:R2→R2 from the uv-plane to the xy-plane is given byx=−4v−3u−1, y=3−4u−5v.(a) Find the absolute value of the determinant of the Jacobian for this change of coordinates.∣∣∣∂(x,y)∂(u,v)∣∣∣=∣∣∣det⎡⎣⎢⎢⎤⎦⎥⎥∣∣∣= (b) If a region D∗ in the uv-plane has area 7.25, find the area of the region T(D∗) in the xy-plane.

Answer 1

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,

$\left[\begin{array}{ccc}dx\\dy\end{array}\right] = \left[\begin{array}{ccc}dx/dv&dx/du\\dy/dv&dy/du\end{array}\right] \left[\begin{array}{ccc}dv\\du\end{array}\right]$

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,

$J = \left[\begin{array}{ccc}-4&-3\\-5&-4\end{array}\right]$ .

Differential area transformation from v-u coordinate system to the x-y coordinate systen is given by,

$dxdy = Det(J)dvdu$

where Det(J), is evaluated as below,

$Det(J) = (-4)*(-4) - (-3)*(-5) = 1$

which we plug into the above equation to obtain,

$dxdy = dvdu$

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,

$Area T(D) = dxdy = dvdu =Area D= 7.25$