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

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

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

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