xyj around the square 6. Use Green's Theorem to calculate the circulation of F 0 < x < 1, 0 < y< 1, oriented counterclockwise.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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6 and 7 please
6. Use Green's Theorem to calculate the circulation of F = xy j around the square
0 < x < 1, 0 < y< 1, oriented counterclockwise.
7. Caleulate (² – y) i + (1² + ») j) • dř i:
· dr
(a) C is the circle (x – 5)² + (y – 4)² = 9 oriented counterclockwise.
(b) C is the circle (x – a)2 + (y – b)² = R² in the xy-plane oriented counterclock-
wise.
Fact: The line integral of F
Green's Theorem, measures the area of the region enclosed by the curve.
xj around a closed curve in the xy-plane, oriented as in
8. Calculate the area of the region within the ellipse x2/a² +y²/b² = 1 parameterized
by x = a cos t, y = b sint for 0<t < 2n. Sketch the curve.
%3D
9. Calculate the area of the region within the folium of Descartes, x + y° =
3t2
1+t3
3xy,
%3D
parameterized by x =
3t
1+t3 , Y =
for 0 <t < oo. Sketch the curve.
Transcribed Image Text:6. Use Green's Theorem to calculate the circulation of F = xy j around the square 0 < x < 1, 0 < y< 1, oriented counterclockwise. 7. Caleulate (² – y) i + (1² + ») j) • dř i: · dr (a) C is the circle (x – 5)² + (y – 4)² = 9 oriented counterclockwise. (b) C is the circle (x – a)2 + (y – b)² = R² in the xy-plane oriented counterclock- wise. Fact: The line integral of F Green's Theorem, measures the area of the region enclosed by the curve. xj around a closed curve in the xy-plane, oriented as in 8. Calculate the area of the region within the ellipse x2/a² +y²/b² = 1 parameterized by x = a cos t, y = b sint for 0<t < 2n. Sketch the curve. %3D 9. Calculate the area of the region within the folium of Descartes, x + y° = 3t2 1+t3 3xy, %3D parameterized by x = 3t 1+t3 , Y = for 0 <t < oo. Sketch the curve.
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