Question texta) Consider the coordinate transformation given by [math: x=2u+3v11]x = {\frac{2\,u+3\,v}{11}},[math: y=u−4v11]y = {\frac{u-4\,v}{11}}. Compute the absolute value of the Jacobian of [math: x,y] with respect to [math: u,v][math: J] = [input] b) Let [math: Ω]\Omega be the second quadrant of the [math: xy]-plane, i.e. the region described by the inequalities [math: x≤0] x \le 0 , [math: y≥0] y \ge 0 . Using the coordinate transform from part a evaluate[math: ∫∫Ωe−t(x−2y)2dxdy] \int\int_\Omega {e^ {- t\,{\left(x-2\,y\right)}^2 }} \: \mathrm{d} x \mathrm{d} y = [input] Here [math: t] is a positive constant. Check Question 1

Question

Question texta) Consider the coordinate transformation given by [math: x=2u+3v11]x = {\frac{2\,u+3\,v}{11}},[math: y=u−4v11]y = {\frac{u-4\,v}{11}}. Compute the absolute value of the Jacobian of [math: x,y] with respect to [math: u,v][math: J] = [input] b) Let [math: Ω]\Omega be the second quadrant of the [math: xy]-plane, i.e. the region described by the inequalities [math: x≤0] x \le 0 , [math: y≥0] y \ge 0 . Using the coordinate transform from part a evaluate[math: ∫∫Ωe−t(x−2y)2dxdy] \int\int_\Omega {e^ {- t\,{\left(x-2\,y\right)}^2 }} \: \mathrm{d} x \mathrm{d} y  = [input] Here [math: t] is a positive constant. Check Question 1

BlackTom AI · Accepted Answer

We begin by restating the given coordinate transformation and then compute its Jacobian.

Transformation: x = (2u + 3v)/11 and y = (u − 4v)/11.

Compute partial derivatives:
- ∂x/∂u = 2/11, ∂x/∂v = 3/11
- ∂y/∂u = 1/11, ∂y/∂v = −4/11

Form the Jacobian matrix J_uv = [ [∂x/∂u, ∂x/∂v], [∂y/∂u, ∂y/∂v] ] = [ [2/11, 3/11], [1/11, −4/11] ].

Compute the determinant to obtain the Jacobian:
det(J_uv) = (2/11)*(−4/11) − (3/11)*(1/11) = (−8/121) − (3/121) = −11/121.

The problem asks for the absolute value of the Jacobian of (x, y) with respect to (u, v). Taking the absolute value yields |det(J_uv)| = 11/121.

Therefore, the Jacobian magnitude is 11/121.

Regarding part (b), the task involves evaluating an integral over the second quadrant after applying the transformation. Without explicit instruction or options for this part, we cannot select among possible choices. However, the key foundational step is that the Jacobian magnitude from part (a) is 11/121, which would be used to transform dx dy into |det(J_uv)| du dv in the integral.

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