608 CHAPTER 30. STOKES AND GREEN’S THEOREMS

30.6 Exercises1. Determine whether the vector field(

2xy3 sinz4,3x2y2 sinz4 +1,4x2y3 (cosz4)z3 +1)

is conservative. If it is conservative, find a potential function.

2. Determine whether the vector field(2xy3 sinz+ y2 + z,3x2y2 sinz+2xy,x2y3 cosz+ x

)is conservative. If it is conservative, find a potential function.

3. Determine whether the vector field(2xy3 sinz+ z,3x2y2 sinz+2xy,x2y3 cosz+ x

)is conservative. If it is conservative, find a potential function.

4. Find scalar potentials for the following vector fields if it is possible to do so. If it isnot possible to do so, explain why.

(a)(y2,2xy+ sinz,2z+ ycosz

)(b)

(2z(cos(x2 + y2

))x,2z

(cos(x2 + y2

))y,sin

(x2 + y2

)+2z

)(c) ( f (x) ,g(y) ,h(z))

(d)(xy,z2,y3

)(e)

(z+2 x

x2+y2+1 ,2y

x2+y2+1 ,x+3z2)

5. If a vector field is not conservative on the set U , is it possible the same vector fieldcould be conservative on some subset of U? Explain and give examples if it is possi-ble. If it is not possible also explain why.

6. Prove that if a vector field F has a scalar potential, then it has infinitely many scalarpotentials.

7. Here is a vector field: F ≡(2xy,x2 −5y4,3z2

). Find

∫C F ·dR where C is a curve

which goes from (1,2,3) to (4,−2,1).

8. Here is a vector field: F ≡(2xy,x2 −5y4,3

(cosz3

)z2). Find

∫C F ·dR where C is a

curve which goes from (1,0,1) to (−4,−2,1).

9. Find∫

∂U F ·dR where U is the set {(x,y) : 2 ≤ x ≤ 4,0 ≤ y ≤ x} and

F (x,y) = (xsiny,ysinx)

10. Find∫

∂U F ·dR where U is{(x,y) : 2 ≤ x ≤ 3,0 ≤ y ≤ x2

}and

F (x,y) = (xcosy,y+ x)