   Chapter 16, Problem 31RE

Chapter
Section
Textbook Problem

Verify that Stokes’ Theorem is true for the vector field F(x, y, z) = x2 i + y2 j + z2 k, where S is the part of the paraboloid z = 1 − x2 − y2 that lies above the xy-plane and S has upward orientation.

To determine

To verify: The Stokes’ Theorem is true for the vector field F(x,y,z)=x2i+y2j+z2k .

Explanation

Given data:

F(x,y,z)=x2i+y2j+z2k , z=1x2y2 .

Formula used:

Write the expression for Stokes’ Theorem.

ScurlFdS=CFdr (1)

Write the expression for curlF .

curlF=|ijkxyzPQR| (2)

Consider the expression as follows.

F(x,y,z)=x2i+y2j+z2k (3)

Write the expression for F(x,y,z) .

F(x,y,z)=Pi+Qj+Rk (4)

Compare equations (3) and (4).

P=x2Q=y2R=z2

Substitute x2 for P , y2 for Q and z2 for R in equation (2),

curlF=|ijkxyzx2y2z2|={i[(z2)y(y2)z]j[(z2)x(x2)z]+k[(y2)x(x2)y]}=ij+k=0

Find the value of ScurlFdS .

ScurlFdS=S(0)dS

ScurlFdS=0 (5)

Since, the paraboloid lies above the xy-plane , the value of z=0 .

Consider the expression for part of the paraboloid as follows.

z=1x2y2

Substitute 0 for z ,

0=1x2y2

x2+y2=1 (6)

Write the expression for circle.

x2+y2=r2 (7)

Compare equations (6) and (7)

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