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Chapter 14: Q.4 (page 1140)

Make a chart of all the new notation, definitions, and theorems in this section, including what each new thing means in terms you already understand.

Short Answer

Expert verified

Defination covered :

Stokes' Theoram

Curl

Line intergral

Surface intergral

Step by step solution

01

Step 1. Given 

We need to make a chart .

02

Step 2. Chart.

Allthenewnotationareasfollows:·GeneralizeGreen'stheoremtosmoothsurfaceotherthantheplane.·Stokes'theoram,·∫cF(x,y,z).dr,·∫∫scurlF(x,y,z).ndS.Chart:

NotationMeaning, Definition or Theorems
Stokes' Theoram

Let S be an oriented, smooth or piecewise-smooth surface bounded by a curve C. Suppose that n is an oriented unit normal vector of S and C has a parametrization that traverses C in the counterclockwise direction with respect to n.


If a vector field F = F1i+F₂j+F3k is defined on S, then,

∫cF(x,y,z).dr=∫∫scurlF(x,y,z).ndS.


∫cF(x,y,z).dr
Line intergral of vector field.
∫∫scurlF(x,y,z).ndS.
Surface intergral of curl of vector field .

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Most popular questions from this chapter

Give a smooth parametrization of the upper half of the unit sphere in terms of x and y.

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

Compute dS for your parametrization in Exercise 9.

How would you show that a given vector field in ℝ3 is not conservative?

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 ∫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 ∫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 ∫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 ∫SF(x,y,z).ndS,the direction of the normal vector is irrelevant.

(h) True or False: In computing ∫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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