l_notes18 - 1 Lecture XVIII Change of Variables Vector...

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

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 1 Lecture XVIII Change of Variables; Vector Fields Change of Variables Recall from lecture 17 that we change variables in integrals by the following formula: f ( x, y ) dxdy = f ( x ( u, v ) , y ( u, v )) ˆ R R ∂x ∂x ∂u ∂v ∂y ∂y dudv, ∂u ∂v where R and ˆ R are corresponding regions in the xy and uv planes. Changing variables lets us easily compute integrals. For example, let us find the area in E 2 enclosed by the curves xy = 1 , xy = 3, xy 2 = 1, and xy 2 = 2 in the first quadrant. Denote the region enclosed by these curves by R . If we choose new coordinates u = xy and v = xy 2 , the region ˆ R corresponding to R in the uv plane is the rectangle of vertices (1 , 1) , (3 , 1), (3 , 2), and (1 , 2). Changing variables to u, v , the area of R is given by ∂ ( x, y ) ∂ ( u, v ) dudv = 2 3 dxdy = 2 3 ∂ ( u, v ) ∂ ( x, y ) 1 dudv = R 1 1 1 1 2 3 2 3 1 y x = 1 dudv = dudv = xy 2 y 2 2 xy 1 1 1 1 = 2 3 1 dudv = 2 2 dv = 2 ln 2 ....
View Full Document

This note was uploaded on 09/24/2011 for the course MATH 1802 taught by Professor Duorg during the Two '04 term at Macquarie.

Page1 / 2

l_notes18 - 1 Lecture XVIII Change of Variables Vector...

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

View Full Document Right Arrow Icon
Ask a homework question - tutors are online