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)
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