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16.3 The Fundamental Theorem for Line Integrals

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16.3 The Fundamental Theorem for Line Integrals

8. Since Fx = 2x cos xy−x2y sin xy and Fy = x cos xy−x2y sin xy+

x cos xy, therefore F is conservative.

Z

xy cos xy + sin xy dx = 1

y(xy sin xy) + c = x cos xy + c 14.

F = ( y2

1 + x2, 2y tan−1x) = ∇(y2tan−1x) Z

C

F · T = y2tan−1x |(1,2)(0,0)= 4 ∗ π 4 20.

∂x(4xy − 9x4y2) = ∂y(2y2− 12x3y3) Z (3,2)

(1,1)

2y2− 12x3y3dx + 4xy − 9x4y2dy = 2xy2− 3x4y3 |(3,2)(1,1)= −1919 23. H F · T > 0, F is non-conservative.

24. H F · T = 0, F is conservative.

33. Qx = x2+ y2− 2x2 and Py = −(x2+ y2) + 2y2 Z

x2+y2=1

F · T = Z

(− sin θ, cos θ) · (− sin θ, cos θ) dθ = 2π 34.

∇(− c

| x |) = cx

| x |3 Z

C

−F · T = − Z

d(− c

| x |) = c

| x | |dd2

1= c(1 d2 − 1

d2) Gravitational work = GMm|x| |perihelionaphelion , Electrical work = Qq|x| |

1 2∗10−12 10−12

1

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