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Chapter 14: Q. 57 (page 1134)

Prove that, for any conservative vector field F(x, y),

鈭�R鈥�鈭�F2鈭�x鈭�鈭�F1鈭�ydA=0

for any simply connected region R鈯�R2 whose boundary is smooth or piecewise smooth.

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Most popular questions from this chapter

Integrate the given function over the accompanying surface in Exercises 27鈥�34.f(x,y,z)=ez, where S is the portion of the unit sphere in the first octant.

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Find the flux of the given vector field through a permeable membrane described by surface S.

F(x,y,z)=鈭�zi+yj+xk, where S is the surface with the equation y=coszfor 1鈮�x鈮�5and0鈮�z鈮�蟺2.

Use what you know about averages to propose a formula for the average rate of flux of a vector field F(x, y ,z) through a smooth surface S in the direction of n.

ComputethedivergenceofthevectorfieldsinExercises17鈥�22.G(x,y,z)=yz2i鈭�xsinzj+x2exyk
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