6 and 7 please 6. Use Green's Theorem to calculate the circulation of F = xy j around the square0 < 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 FGreen's Theorem, measures the area of the region enclosed by the curve.xj around a closed curve in the xy-plane, oriented as in8. Calculate the area of the region within the ellipse x2/a² +y²/b² = 1 parameterizedby x = a cos t, y = b sint for 0

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Answered: xyj around the square 6. Use Green's…

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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Let F = (-z2, 2zx, 4y - x2}, and let C be a simple closed curve in the plane x + y + z = 4 that encloses a region of area 16 (Figure 20). Calculate ∮C F • dr, where C is oriented in the counterclockwise direction (when viewed from above the plane).
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Verify Stokes' Theorum for F(x,y,z) = (y,-x,yz) if S is the paraboloid z = x2 + y2 with x2 + y2 = 1, z=1 as its boundary.
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5.Consider the ellipsoid V(x,y,z)=rx2+σy2+σ(z−2r)2=c>0.Vx,y,z=rx2+σy2+σz−2r2=c>0. a.Calculate dVdtdVdt along trajectories of the Lorenz equations (1).
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The outflow of the field F(x,y,z)=(e^y−xz^2,x−yx^2,y−zy^2), across the edge of the solid contained in the semispace x≥0 and inside the sphere of center (0,0,0) and radius 5 is ?

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