Change
of Variables in Integrals
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Consider regions, R, in
the xy-plane that are parallelograms. Let dx dy be the original element of area for region R and
JT(u,v)
du dv be the rectangular element of area of the transformed region, S, as shown in the
figure below. Then given sufficient
continuity dx
dy = | JT(u,v) | du dv where = JT(u,v) = the Jacobian
transformation defined as follows: ∂x/∂u ∂x∂v JT(u,v) = det where | JT(u,v)
| is the magnitude of the Jacobian ∂y/∂u ∂y∂v
and where det denotes the magnitude of the 2 x 2
determinant of the partial derivatives. The integral then becomes: I = ∫ ∫ F(x,y) dx dy = ∫ ∫ G(u,v) | JT(u,v) | du dv R S R is the original region
of integration and S is the transformed region of integration. Click here to continue
with discussion. |
Copyright © 2017 Richard C. Coddington
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