Math Problem Statement

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

Solution

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Math Problem Analysis

Mathematical Concepts

Multivariable Calculus
Jacobian Determinant
Double Integrals
Coordinate Transformations

Formulas

Jacobian Matrix: J = [∂x/∂u ∂x/∂v; ∂y/∂u ∂y/∂v]
Jacobian Determinant: det(J) = (∂x/∂u)(∂y/∂v) - (∂x/∂v)(∂y/∂u)
Double Integral: ∫∫_D f(x² + y²) du dv

Theorems

Change of Variables Theorem
Jacobian Determinant for Coordinate Transformations

Suitable Grade Level

Undergraduate