(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Suppose f is a continously differentiable real valued function on R^3 and F is a continously differentiable vector field

Prove 1)##\oint (f \nabla g +g\nabla f) \cdot dr=0##

2) ##\oint(f \nabla f)\cdot dr=0##

2. Relevant equations

##\nabla f = f_z i+ f_y j+f_z k##

Real valued function ##f(x,y,z)## and ##g(x,y,z)##

3. The attempt at a solution

1)

##f \nabla g =fg_x i +fg_y j+fg_z k##

##g \nabla f =gf_x i +gf_y j+gf_z k##

##\implies (f \nabla g + g \nabla f )\cdot dr##

##= (fg_x i +fg_y j+fg_z k+gf_x i +gf_y j+gf_z k)\cdot(dx i+dyj+dzk)##

2)

##(f \nabla f)\cdot dr= (ff_xi+ff_yj+ff_zk)\cdot(dxi+dyj+dzk)##

How do these work out to be 0?

Thanks

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# Homework Help: Calculus Identities

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