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Chapter 14: Q. 6 (page 1119)

Relate the integrand $$F \cdot n dS$$ to the discussions of work in Sections 6.4 and 14.2.

Short Answer

Expert verified

$$W=\int F \cdot n dS$$

Step by step solution

01

Step 1. Given Information

Integrand $$F \cdot n dS$$

Work in Sections 6.4 and 14.2

02

Step 2. Explanation

Work is given as the dot prioduct of Force and dIsplacement.

$$\implies W=F \cdot \bigtriangleup x$$

The expression for small amount of work done is given as,

$$dW=F \cdot dx$$

Hence, Integrating the above expression, we get

$$W=\int F \cdot n dS$$

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Most popular questions from this chapter

Compute dS for your parametrization in Exercise 7.

Find

∫S F(x,y,z)⋅ndSifF(x,y,z)=2xzi+2yzj−18k

and S is the portion of the hyperboloid x2+y2-9=z2that lies between the planes

z = −4 and z = 0, with n pointing outwards.

F(x,y,z)=zcosyzj+zsinyzk, where S is the portion of the plane with equation 2x−8y−10z=42 that lies on the positive side of the rectangle with corners0,-π,0(0,π,0),(0,π,π),(0,−π,π)in theyz-plane.

Why do surface integrals of multivariate functions not include an n term, whereas surface integrals of vector fields do include this term?

ComputethedivergenceofthevectorfieldsinExercises17–22.F(x,y)=yx2i−xcosyj
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