20E_sheet2

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

This preview shows pages 1–2. Sign up to view the full content.

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 (

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online