week4 - 4 4.1 Multiple integrals; vectors Multiple...

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4 Multiple integrals; vectors 4.1 Multiple integrals Let’s review this subject by doing various examples of integrating a function f ( x,y ) over a region of 2-space: Ex. 1: I = Z region y xdydx = Z 4 0 dx x Z x 0 y dy = Z 4 0 dx x · 1 2 x = 32 5 (1) x y y=x 1/2 2 4 Figure 1: Ex. 1 Q: how about changing the order of integration? We could do the x -integral first obviously. But we need to be careful of the order of the limits.: I = Z 2 y =0 dy y Z 4 x = y 2 xdx = 32 5 . (2) Q: Why change the order? Sometimes it’s important: Ex. 2: I = Z ln16 x =0 dx Z 4 y = e x/ 2 dy ln y (3) Can’t do R dy/ ln y , so switch: (4) 1
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x y y=e x/2 4 Figure 2: Ex. 1 So let’s draw a new figure (always draw a figure if you switch!). I = Z 4 y =1 dy ln y Z 2ln y 0 dx = Z 4 y =1 dy · 2 = 6 (5) 4.2 Change of variables: the Jacobian First, let’s do a standard example where we don’t get into formalities: x y dr rd θ 1 1 d θ y= 1-x 2 Figure 3: Ex. 1 Ex. 3: I = Z 1 x =0 Z 1 - x 2 y =0 e - ( x 2 + y 2 ) dydx (6) 2
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This will certainly be easier in polar coordinates x = r cos θ , y = r sin θ , I = Z π/ 2 θ =0 Z 1 r =0 e - r 2 rdrdθ = π 4 (1 - 1 /e ) . (7) Note the measure for the integral in polar coords., dxdy = r dr dθ, (8) is just the size of the little area element in Fig. 3, but can be obtained formally. In general the change of variables in 2D is defined Z Z R f ( u,v ) dudv = Z Z R 0 f ( u ( r,s ) ,v ( r,s )) · J ± u,v r,s ¶fl dr ds, (9) where R is a region in 2D and R 0 is the transformed region, which need not have the same shape. J is the “Jacobian of transformation” J ± u,v r,s = ∂u ∂r ∂u ∂s ∂v ∂r ∂v ∂s = 1 J r,s u,v · , (10) i.e. the determinant of the partial derivatives as shown. If you have forgotten
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week4 - 4 4.1 Multiple integrals; vectors Multiple...

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