Question

Suggest a substitution or transformation that will simplify the following integrands, and find their Jacobians. Use u and v as the substitution variables.

(a) ∬ᵣ (4x + 2y) sin(x - 3y) dA
Start with
u = (4x + 2y)
v = (x - 3y)
Give the substitutions for x and y:
x =
y =
Find the absolute value of the Jacobian

Answer 1

To simplify the given integrand, we can make the substitutions x = (u - 2y) / 2 and y = (u - 2v) / 10. This changes the integrand to (u + 4y) sin(u - 4y) (1/10) du dv. The Jacobian for this transformation is 1/10.

Step-by-step explanation:

To simplify the given integrand of ∬ᵣ (4x + 2y) sin(x - 3y) dA, we can make the substitution u = 4x + 2y and v = x - 3y. To find the substitutions for x and y, we can solve the two equations simultaneously:

4x + 2y = u → 2x = u - 2y → x = (u - 2y) / 2

x - 3y = v → (u - 2y) / 2 - 3y = v → u - 4y - 6y = 2v → -10y = 2v - u

Simplifying the above equation, we get y = (u - 2v) / 10. Substituting this value of y back into x = (u - 2y) / 2, we can rewrite the integrand in terms of u and v:

∫∫ (4x + 2y) sin(x - 3y) dA = ∫∫ (u + 4y) sin(u - 4y) (1/10) du dv.

The Jacobian for this transformation can be found by taking the determinant of the Jacobian matrix, which is given by J = du/dx * dv/dy - dv/dx * du/dy. In this case, we have:

J = (2/2) * (1/10) - (0/2) * (-2/10) = 1/10.