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Calculus Identities 
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#1
May1212, 01:40 PM

P: 652

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 


#2
May1212, 02:53 PM

P: 435

Any chance ## f \nabla g +g\nabla f ## and ## f \nabla f## might be conservative?



#3
May1212, 02:54 PM

PF Gold
P: 836

In my opinion, there are some missing information in your problem.
Is ##g## also a continously differentiable real valued function in R^3? 


#4
May1212, 04:13 PM

P: 652

Calculus Identities
##...f_xi...## It does not mention anything about g but perhaps we take it that it is also a real valued function? I believe I left out the following important information C is a smooth, simple closed curve which lies on the surface of a paraboloid in R^3. I guess this means integrand is conservative, right? But I still not sure how it goes to 0, there must be additional lines Thanks 


#5
May1212, 05:40 PM

PF Gold
P: 836




#6
May1312, 03:07 AM

P: 652

Suppose that sigma and C satisfy the hypothesis of Stokes Theorem and that f and g have continous second order partial dervivatives. Prove each of the following ##\oint_C (f \nabla g) \cdot dr = \oint \oint_\sigma (\nabla f \times \nabla g)\cdot dS## ##\oint_C (f \nabla f) \cdot dr=0## ##\oint (f \nabla g +g \nabla f)\cdot dr=0## I am interested in the last 2 but maybe the first one allows me to complete the last 2? Thanks 


#7
May1312, 08:01 AM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,348




#8
May1312, 09:03 AM

P: 652

THanks 


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