Consider the transformation T:x=(41/40)u−(9/41)v , y=(9/41)u+(40/41)v
A. Computer the Jacobian:
delta(x,y)/delta(u,v)= ?
B. The transformation is linear, which implies that it transforms lines into lines. Thus, it transforms the square S:−41<=u<=41, −41<=v<=41 into a square T(S) with vertices:
T(41, 41) = ( ? , ? )
T(-41, 41) = ( ? , ? )
T(-41, -41) = ( ? , ? )
T(41, -41) = ( ? , ? )
C. Use the transformation T to evaluate the integral double integral_T(S) (x^2+y^2) dA
A. To compute the Jacobian of the given transformation T, we need to calculate the partial derivatives of x and y with respect to u and v.
The Jacobian matrix J is given by:
J = [ (delta x)/(delta u) (delta x)/(delta v) ]
[ (delta y)/(delta u) (delta y)/(delta v) ]
Let's calculate the partial derivatives:
(delta x)/(delta u) = (41/40)
(delta x)/(delta v) = -(9/41)
(delta y)/(delta u) = (9/41)
(delta y)/(delta v) = (40/41)
Now we can form the Jacobian matrix:
J = [ (41/40) -(9/41) ]
[ (9/41) (40/41) ]
B. The given transformation T is linear, which means it transforms lines into lines. To find the vertices of the transformed square T(S), we substitute the corners of the square S into the transformation equations.
Substituting (-41, -41) into the transformation equations:
T(-41, -41) = ( (41/40)(-41) - (9/41)(-41), (9/41)(-41) + (40/41)(-41) )
= (-41/40 - 9/41, -9/41 + (-40/41))
= (-1681/1640, -49/41)
Substituting (-41, 41) into the transformation equations:
T(-41, 41) = ( (41/40)(-41) - (9/41)(41), (9/41)(-41) + (40/41)(41) )
= (-41/40 - 9/41, -9/41 + 40/41)
= (-1681/1640, 31/41)
Substituting (41, -41) into the transformation equations:
T(41, -41) = ( (41/40)(41) - (9/41)(-41), (9/41)(41) + (40/41)(-41) )
= (41/40 + 9/41, 9/41 - 40/41)
= (1681/1640, -49/41)
Substituting (41, 41) into the transformation equations:
T(41, 41) = ( (41/40)(41) - (9/41)(41), (9/41)(41) + (40/41)(41) )
= (41/40 + 9/41, 9/41 + 40/41)
= (1681/1640, 31/41)
Thus, the vertices of the transformed square T(S) are:
T(41, 41) = (1681/1640, 31/41)
T(-41, 41) = (-1681/1640, 31/41)
T(-41, -41) = (-1681/1640, -49/41)
T(41, -41) = (1681/1640, -49/41)
C. To evaluate the given integral, we need to calculate the double integral of (x^2 + y^2) over the transformed square T(S). This is equivalent to finding the area under the curve defined by (x^2 + y^2) within the transformed square.
The integral can be expressed as:
double integral_T(S) (x^2 + y^2) dA
Integrating over T(S), we substitute the transformed coordinates (x, y) into the expression (x^2 + y^2):
double integral_T(S) (x^2 + y^2) dA = double integral_T(S) ((41/40*u - 9/41*v)^2 + (9/41*u + 40/41*v)^2) dA
Substituting the transformation equations x and y in terms of u and v, we have:
double integral_T(S) ((41/40*u - 9/41*v)^2 + (9/41*u + 40/41*v)^2) dA = double integral_T(S) ((41/40*u - 9/41*v)^2 + (9/41*u + 40/41*v)^2) |J| du dv
where |J| is the determinant of the Jacobian matrix, which in this case is |J| = (41/40)*(40/41) - (-(9/41))*(9/41) = 1.
Therefore, the integral simplifies to:
double integral_T(S) (x^2 + y^2) dA = double integral_T(S) ((41/40*u - 9/41*v)^2 + (9/41*u + 40/41*v)^2) du dv
Now you can compute this integral over the transformed square T(S) using appropriate integration techniques.