   Chapter 16.8, Problem 1E

Chapter
Section
Textbook Problem

1. A hemisphere H and a portion P of a paraboloid are shown. Suppose F is a vector field on ℝ3 whose components have continuous partial derivatives. Explain why ∬ H curl   F  ⋅   d S   =   ∬ P curl   F   ⋅   d S To determine

To explain: The reason for satisfying the expression HcurlFdS=PcurlFdS .

Explanation

Given data:

Consider the hemisphere H and a portion P of a paraboloid are shown in the question.

Both hemisphere H and a portion P of a paraboloid are oriented piecewise-smooth surfaces, which are bounded by simple, closed, and smooth curve x2+y2=4 , z=0 .

This means Both surfaces have the same boundary, which is the circle x2+y2=4 .

The hemisphere H and portion P satisfy the hypotheses of Stokes’ theorem since both surfaces are oriented surfaces with the same oriented boundary curve C. Therefore, by Stokes’ theorem, write the following expression,

HcurlFdS=P

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