   Chapter 16, Problem 9RQ

Chapter
Section
Textbook Problem

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If F and G are vector fields, then curl(F + G) = curl F + curl G

To determine

Whether the statement “If F and G are vector fields, then curl(F+G)=curlF+curlG ” is true or false.

Explanation

Formula used:

Write the expression for curl.

curlF=|ijkxyzF1F2F3| (1)

Consider the two different three-dimensional vector fields as follows.

F(x,y,z)=F1i+F2j+F3kG(x,y,z)=G1i+G2j+G3k

Find the value of curl(F+G) .

curl(F+G)=|ijkxyzF1+G1F2+G2F3+G3|

curl(F+G)={(y(F3+G3)z(F2+G2))i(x(F3+G3)z(F1+G1))j+(x(F2+G2)y(F1+G1))k} (2)

Find the value of curlF .

curlF=|ijkxyzF1F2F3|=(yF3zF2)i(xF3zF1)j+(xF2yF2)k

Modify equation (1) as follows.

curl(G)=|ijkxyzG1G2G3|=(yG3zG2)i(xG3zG1)j+(xG2yG2)k

Find the value of curlF+curlG

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