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As mentioned the
challenging part in changing variables from (x,y)
to (u,v) is to find the
transformation x = x(u,v) and y = y(u,v) from the region R in the xy-plane
to region S in
the uv-plane.
Then use this information to
calculate the Jacobian transformation, JT(u,v).
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i.e. Use the Jacobian,
JT(u,v), to rewrite the transformed
integral as follows:
I
= ∫ ∫ F(x,y) dA = ∫ ∫ G(u,v) | JT(u,v) | du dv
R S
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The table below details
one strategy to find the Jacobian transformation
from parallelogram
regions, R, in the xy-plane to rectangular regions, S, in the uv-plane.
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Step 1 |
Graph the region, R,
defined by:
y ˗ f1(x) = C1, y ˗ f1(x) = C2 ,
by
y ˗ f2(x) = D1, y ˗ f2(x) = D2 .
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Step 2 |
Determine the
coordinates of the vertices for each corner of the region
in R and the equations
describing each "side" of region R.
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Step 3 |
Find the new variables,
u and v, by setting u = y ˗ f1(x) and
v = y ˗ f2(x)
where C1 ≤ u
≤ C2
and D1 ≤
v ≤ D2.
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Step 4 |
The coordinates of the
vertices for each corner of the region in S are
(C1 , D1), (C1 , D2), (C2 , D1), and
(C2 , D2) .
Plot the rectangle, S.
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Step 5 |
The limits of
integration in the u-v plane are C1
≤ u ≤ C2 and
D1 ≤ v
≤ D2 .
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Step 6 |
Calculate the Jacobian, JT(u,v)
by taking the partial derivatives
∂x/∂u, ∂x/∂v, ∂y/∂u, and ∂y/∂v .
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Step 7 |
Use the Jacobian to evaluate the integral, I,
D2 C2
I
= ∫ ∫ F(x,y) dA = ∫ ∫ G(u,v) | JT(u,v)
| du dv
R D1 C1
S
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Click here for an example.
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