- #1
bugatti79
- 794
- 1
Homework Statement
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##
Homework Equations
##\nabla f = f_z i+ f_y j+f_z k##
Real valued function ##f(x,y,z)## and ##g(x,y,z)##
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