MIT18 02SC Pb 54 Comb

Problems: Polar Coordinates and the Jacobian 1. Let r = x . −1 y ∂ (r, θ) 1 2 + y 2 and θ = tan . Directly calculate the Jacobian = ∂ (x, y) x r Answer: Because we are familiar with the change of variables from rectangular to polar ∂ (r, θ) ∂ (x, y) coordinates and we know that · = 1, this result should not come as a surprise. ∂ (x, y) ∂ (r, θ) ∂ (r, θ) r x r y = θ x θ y ∂ (x, y) 2x 2y √ √ 2 2 2 x 2 +y 2 x 2 +y −y/x 2 1/x = 1+(y/x) 2 x √ y √ 1+(y/x) 2 2 2 x 2 +y x 2 +y = − y 2 x 2 +y 2 2 +y x x x y r r = − y x 2 2 r r x 2 + y 2 1 = = . 3 r r √ For the change of variables x = u, y = r 2 − u 2 , write dx dy in terms of u and r. 2. ∂ (x, y) Answer: We know dx dy = du dr. ∂ (u, r) ∂ (x, y) x u x r = ∂ (u, r) y u y r 1 0 −u √ √ r r 2 −u 2 r 2 −u 2 r = = √ 2 − u 2 r r Hence dx dy = √ du dr. r 2 − u 2

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Transcript of MIT18 02SC Pb 54 Comb

Problems: Polar Coordinates and the Jacobian 1. Let r=x. 1y (r,) 12+y2and =tan . Directly calculate the Jacobian =(x,y)x r Answer: Because we are familiar with the change of variables from rectangular to polar(r,) (x,y)coordinatesandweknowthat =1,thisresultshouldnotcomeasasurprise.(x,y) (r,)

(r,) rxry= xy (x,y)

2x 2y 2 22 x2+y 2 x2+yy/x21/x=

1+(y/x)2xy1+(y/x)2

2 2x2+y x2+y= y2x2+y2 2+y x x

x yr r= y x2 2r rx2+y21 = = .3r r

For the change of variables x=u, y= r2u

2, write dxdy in terms of u and r. 2. (x,y) Answer: We know dxdy=dudr. (u,r)

(x,y) xuxr= (u,r) yuyr

1 0 u rr2u2r2u2r= = 2u2rrHence dxdy= dudr.r2u2MIT OpenCourseWarehttp://ocw.mit.edu18.02SC Multivariable CalculusFall 2010For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.