By a change of variable x(u,v) = uv, y(u,v) = v/u is double integral, the integrand f(x,y) change to f(uv, v/u)φ(u,v). Thenφ (u,v) isa)2 u/vb)2uvc)v2d)1Correct answer is option 'A'. Can you explain this answer?

Question

By a change of variable x(u,v) = uv, y(u,v) = v/u is double integral, the integrand f(x,y) change to f(uv, v/u)&phi;(u,v). Then&phi; (u,v) isa)2 u/vb)2uvc)v2d)1Correct answer is option 'A'. Can you explain this answer?

Answer

Let's consider the change of variables from (x, y) to (u, v), given by x(u, v) = uv and y(u, v) = v/u.

To find the Jacobian of this transformation, we need to find the partial derivatives of x with respect to u and v, and the partial derivatives of y with respect to u and v:

∂x/∂u = v
∂x/∂v = u
∂y/∂u = -v/u^2
∂y/∂v = 1/u

The Jacobian of the transformation is then given by the determinant of the matrix:

J = |∂x/∂u ∂x/∂v|
|∂y/∂u ∂y/∂v|

J = |v u|
|-v/u^2 1/u|

The determinant of this matrix is J = v/u + v/u = 2v/u.

Now, to perform the change of variables in the double integral, we substitute x = uv and y = v/u into the integrand f(x, y):

f(x, y) → f(uv, v/u)

Finally, we need to account for the change in area element when transforming from (x, y) to (u, v). The area element dA in (x, y) coordinates is given by dA = dx dy, and in (u, v) coordinates it is given by dA' = |J| du dv.

Thus, the double integral with the change of variables is:

∫∫ f(uv, v/u) |J| du dv

Answer

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