Consider the region DC R defined by the inequalities a? + y? > 1, x² + y? < 4, y > 0, and y
Consider the region DC R defined by the inequalities a? + y? > 1, x² + y? < 4, y > 0, and y
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Consider the region DCR² defined by the inequalities x? + y? > 1, x² + y² < 4, y > 0, and y < r. Let w be the two-form w = (x + 2) dx A dy. In this equation we evaluate the integral of w over the
region D with canonical orientation using polar coordinates.
Let ø : D2 → D be the change of coordinates from Cartesian to polar coordinates:
$(r, t) = (r cos(t),r sin(t)).
The map ø is bijective if D2 is the following rectangular region in the (r, t)-plane:
D2 = {(r, t) E R² | r E[|
], t €[
1}.
The transformation o is orientation-preserving (the determinant of its Jacobian is positive), and so we know that the integral of w over D is equal to the integral of the pullback o*w over D2. We calculate
the pullback two-form:
O*w =
dr A dt.
Finally we can evaluate the integral of o*w over D2 using double integrals. We get:
W =
$*w =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0fbbb112-1902-4490-9475-b2c99ad6e439%2Fa62ab93b-e8e4-4f40-b33e-292816277439%2Fbyxsgd_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the region DCR² defined by the inequalities x? + y? > 1, x² + y² < 4, y > 0, and y < r. Let w be the two-form w = (x + 2) dx A dy. In this equation we evaluate the integral of w over the
region D with canonical orientation using polar coordinates.
Let ø : D2 → D be the change of coordinates from Cartesian to polar coordinates:
$(r, t) = (r cos(t),r sin(t)).
The map ø is bijective if D2 is the following rectangular region in the (r, t)-plane:
D2 = {(r, t) E R² | r E[|
], t €[
1}.
The transformation o is orientation-preserving (the determinant of its Jacobian is positive), and so we know that the integral of w over D is equal to the integral of the pullback o*w over D2. We calculate
the pullback two-form:
O*w =
dr A dt.
Finally we can evaluate the integral of o*w over D2 using double integrals. We get:
W =
$*w =
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