   Chapter 16.9, Problem 27E

Chapter
Section
Textbook Problem

Prove each identity, assuming that S and E satisfy the conditions of the Divergence Theorem and the scalar functions and components of the vector fields have continuous second-order partial derivatives.27. ∬ S curl   F   ⋅   d S   =   0

To determine

To prove: The expression ScurlFdS=0 .

Explanation

Formula used:

Write the expression for ScurlFdS .

ScurlFdS=Ediv(curlF)dV (1)

Here,

E is the solid region.

Write the expression to find divergence of vector field F(x,y,z)=Pi+Qj+Rk .

divF=xP+yQ+zR (2)

Write the expression to find curl of vector field F(x,y,z)=k1i+k2j+k3k .

curlF=|ijkxyzk1k2k3| (3)

Consider the vector filed F(x,y,z) is ai+bj+ck .

Calculation of curlF :

Substitute a for k1 , b for k2 , and c for k3 in equation (3),

curlF=|ijkxyzabc|=|yzbc|i|xzac|j+|xyab|k=(cybz)i(cxaz)j+(bxay)k=(cybz)i+

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