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lec_week8 - MIT OpenCourseWare http/ocw.mit.edu 18.02...

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MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable Calculus Fall 200 7 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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�� �� 18.02 Lecture 18. Tue, Oct 23, 2007 Change of variables. Example 1: area of ellipse with semiaxes a and b : setting u = x/a , v = y/b , �� �� �� dx dy = ab du dv = ab du dv = πab. ( x/a ) 2 +( y/b ) 2 < 1 u 2 + v 2 < 1 u 2 + v 2 < 1 (substitution works here as in 1-variable calculus: du = 1 dx , dv = 1 dy , so du dv = 1 dx dy . a b ab In general, must find out the scale factor (ratio between du dv and dx dy )? Example 2: say we set u = 3 x 2 y , v = x + y to simplify either integrand or bounds of integration. What is the relation between dA = dx dy and dA = du dv ? (area elements in xy - and uv -planes). Answer: consider a small rectangle of area Δ A = Δ x Δ y , it becomes in uv -coordinates a paral- lelogram of area Δ A . Here the answer is independent of which rectangle we take, so we can take e.g. the unit square in xy -coordinates. In the uv -plane, u = 3 2 x , so this becomes a parallelogram with sides given by 1 1 � � v y 3 1 2 1 3 2 1 vectors 3 , 1 and �− 2 , 1 (picture drawn), and area = det = = 5 = . 1 For any rectangle Δ A = A , in the limit dA = 5 dA , i.e. du dv = 5 dx dy . So . . . dx dy = . . . 1 du dv. 5 General case: approximation formula Δ u u x Δ y , Δ v v x Δ u u x u y Δ x . Δ v v x v y Δ y A small xy -rectangle is approx. a parallelogram in uv -coords, but scale factor depends on x and y Δ x + u y � � Δ x + v y Δ y , i.e. now. By the same argument as before, the scale factor is the determinant. ( u, v ) u x u y Definition: the Jacobian is J . Then du dv = J dx dy. = = | | ( x, y ) v x v y (absolute value because area is the absolute value of the determinant). Example 1: polar coordinates x = r cos θ , y = r sin θ : ( x, y ) = r cos 2 θ + r sin 2 θ = r. cos θ r sin θ r cos θ x r x θ = = sin θ ( r, θ ) y r y θ So dx dy = r dr , as seen before. Example 2: compute 0 1 0 1 x 2 y dx dy by changing to u = x , v = xy (usually motivation is to simplify either integrand or region; here neither happens, but we just illustrate the general method).
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