   Chapter 16, Problem 10RQ

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 curI(F · G) = curl F · curl G

To determine

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

Explanation

Consider the statement as follows.

curl(FG)=curlFcurlG (1)

Consider the standard equation of an curl F for F=Pi+Qj+Rk .

curlF=|ijkxyzPQR| (2)

Consider the two vector field as follows.

F=xi+yj+zkG=yi+xj+zk

Find the value of curlF .

Substitute x for P , y for Q and z for R in equation (2),

curlF=|ijkxyzxyz|={[y(z)z(y)]i[x(z)z(x)]j+[x(y)y(x)]k}={ij+k}=0

Find the value of curlG .

Modify equation (2) as follows

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