Consider the transformation from Cartesian coordinates $(x,y)$ to parametric $(u,v)$ defined by the following: x = u³-3uv² y = 3u-v-v3 Calculate the Jacobian determinent $ J = \frac{\partial(x,y)}{\partial(u,v)) $, then evaluate the double integral over the region $D$ in the $uv$ plane definite by $u^2 + v^2 = 1$: 1= f (x²+y²) du dv D

### Step 1: Calculate the Jacobian Determinant $ J = \frac{\partial(x, y)}{\partial(u, v)} $
Given the transformation:
\[
x = u^3 - 3uv^2, \quad y = 3u - v - v^3,
\]
the **Jacobian matrix** is:
\[
J = 
\begin{pmatrix}
\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\
\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}
\end{pmatrix}.
\]

#### Compute the partial derivatives:
1. **Partial derivatives of $x$:**
   \[
   \frac{\partial x}{\partial u} = \frac{\partial}{\partial u}(u^3 - 3uv^2) = 3u^2 - 3v^2,
   \]
   \[
   \frac{\partial x}{\partial v} = \frac{\partial}{\partial v}(u^3 - 3uv^2) = -6uv.
   \]

2. **Partial derivatives of $y$:**
   \[
   \frac{\partial y}{\partial u} = \frac{\partial}{\partial u}(3u - v - v^3) = 3,
   \]
   \[
   \frac{\partial y}{\partial v} = \frac{\partial}{\partial v}(3u - v - v^3) = -1 - 3v^2.
   \]

Thus, the **Jacobian matrix** is:
\[
J = 
\begin{pmatrix}
3u^2 - 3v^2 & -6uv \\
3 & -1 - 3v^2
\end{pmatrix}.
\]

#### Calculate the Jacobian Determinant:
\[
\det(J) = (3u^2 - 3v^2)(-1 - 3v^2) - (-6uv)(3).
\]
Expanding:
\[
\det(J) = -3u^2 - 9u^2v^2 + 3v^2 + 9v^4 + 18uv = 3v^2 - 3u^2 + 9v^4 - 9u^2v^2 + 18uv.
\]

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### Step 2: Set Up the Double Integral
The integral is:
\[
\iint_D f(x^2 + y^2) \, du \, dv,
\]
where $D$ is the region defined by $u^2 + v^2 = 1$. 

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Consider the transformation from Cartesian coordinates (x,y) to parametric (u,v) defined by the following: x = u³ - 3uv², y = 3u - v - v³. Calculate the Jacobian determinant J = ∂(x,y)/∂(u,v), then evaluate the double integral over the region D in the uv plane defined by u² + v² = 1: ∫∫_D f(x² + y²) du dv.

This problem explores calculating the Jacobian determinant for the transformation from Cartesian to parametric coordinates (u,v), followed by evaluating a double integral over a circular region. Concepts covered include Jacobian matrices, coordinate transformations, and multivariable calculus methods.

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Solution

Step 1: Calculate the Jacobian Determinant $J = \frac{\partial(x, y)}{\partial(u, v)}$

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