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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

Q. Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) A smooth surface with a smooth boundary.

(b) A surface that is not smooth, but that has a smooth boundary.

(c) A surface that is smooth, but does not have a smooth boundary

Give an example of a field with positive divergence at (1, 0, 蟺).

Find the work done by the vector field

F(x,y)=cosx2+4xy2i+2y-4x2yj

in moving an object around the unit circle, starting and ending at (1,0).

Why is dAin Green鈥檚 Theorem replaced by dSin Stokes鈥� Theorem?

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The result of integrating a vector field over a surface is a vector.

(b) True or False: The result of integrating a function over a surface is a scalar.

(c) True or False: For a region R in thexy-plane,dS=dA.

(d) True or False: In computing 鈭�Sf(x,y,z)dS, the direction of the normal vector is irrelevant.

(e) True or False: If f (x, y, z) is defined on an open region containing a smooth surface S, then 鈭�Sf(x,y,z)dSmeasures the flow through S in the positive z direction determined by f (x, y, z).

(f) True or False: If F(x, y, z) is defined on an open region containing a smooth surface S , then 鈭�SF(x,y,z).ndSmeasures the flow through S in the direction of n determined by the field F(x, y, z).

(g) True or False: In computing 鈭�SF(x,y,z).ndS,the direction of the normal vector is irrelevant.

(h) True or False: In computing 鈭�SF(x,y,z).ndS,with n pointing in the correct direction, we could use a scalar multiple of n, since the length will cancel in the dSterm.
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