Math Problem Statement

Let 𝐼=∫∫(𝑥2−𝑦2)𝑑𝑥𝑑𝑦 I

∫ ∫ D ( x 2 − y 2 ) d x d y , where ={(𝑥,𝑦):2≤𝑥𝑦≤3,0≤𝑥−𝑦≤4,𝑥≥0,𝑦≥0} D

{ ( x , y ) : 2 ≤ x y ≤ 3 , 0 ≤ x − y ≤ 4 , x ≥ 0 , y ≥ 0 } Show that the mapping 𝑢=𝑥𝑦 u

x y , 𝑣=𝑥−𝑦 v

x − y maps  D to the rectangle =[2,3]×[0,4] R

[ 2 , 3 ] × [ 0 , 4 ] .

(𝐚) ( a ) Compute ∂(𝑥,𝑦)/∂(𝑢,𝑣) ∂ ( x , y ) / ∂ ( u , v ) by first computing ∂(𝑢,𝑣)/∂(𝑥,𝑦) ∂ ( u , v ) / ∂ ( x , y ) .

(𝐛) ( b ) Use the Change of Variables Formula to show that 𝐼 I is equal to the integral of 𝑓(𝑢,𝑣)=𝑣 f ( u , v )

v over  R and evaluate.

Solution

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Math Problem Analysis

Mathematical Concepts

Double Integrals
Change of Variables
Jacobian Determinants
Integration in Multivariable Calculus

Formulas

Jacobian Determinant: ∂(u, v)/∂(x, y) = |∂u/∂x ∂u/∂y; ∂v/∂x ∂v/∂y|
Change of Variables Formula: ∫∫_D f(x, y) dx dy = ∫∫_R f(u, v) |∂(x, y)/∂(u, v)| du dv

Theorems

Change of Variables Theorem

Suitable Grade Level

Undergraduate Mathematics (Calculus III or Advanced Multivariable Calculus)