20E_sheet2

20E_sheet2 - FACT SHEET FOR 20E EXAM 2 1. From Chapter 6 If...

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FACT SHEET FOR 20E EXAM 2 1. From Chapter 6 If ( x, y )= ± Φ( u, v ) is a map between two domains Ω and ± Φ(Ω) in R 2 , then the change of variables formula for integrals is: ±± Ω f ( ± Φ( u, v ) ) ² ² ² ± Φ ( u, v ) ² ² ² dudv = ± Φ(Ω) f ( x, y ) dxdy . Here ² ² ² ± Φ ( u,v ) ² ² ² is the absolute value of the determinant of the matrix of Frst partial derivatives: ² ² ² ± Φ ( u, v ) ² ² ² = ² ² u x∂ v y - v x∂ u y ² ² . We are writing ( x ( u, v ) ,y ( u, v ) ) for the components of ± Φ. There are two very important particular cases of the above formula. The Frst is polar coordinates: x = r cos( θ ) = r sin( θ ) , where ² ² ² ( x,y ) ( r,θ ) ² ² ² = rdrdθ . ±or example: x 2 + y 2 ± R 2 f ( x, y ) dxdy = ± 2 π 0 ± R 0 f ( r cos( θ ) ,r sin( θ ) ) rdrdθ . The other main important case is the analog of the above formula in three dimensions: x = ρ sin( φ ) cos( θ ) = ρ sin( φ ) sin( θ ) ,z = ρ cos( φ ) , where ² ² ² ( x,y,z ) ( ρ,φ,θ ) ² ² ² = ρ 2 sin( φ ) dρdφdθ . ±or example: x 2 + y 2 + z 2 ± R 2 f (
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20E_sheet2 - FACT SHEET FOR 20E EXAM 2 1. From Chapter 6 If...

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