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Calculus Identities

by bugatti79
Tags: calculus, identities
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bugatti79
#1
May12-12, 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
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gopher_p
#2
May12-12, 02:53 PM
P: 435
Any chance ## f \nabla g +g\nabla f ## and ## f \nabla f## might be conservative?
sharks
#3
May12-12, 02:54 PM
PF Gold
sharks's Avatar
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?

Quote Quote by bugatti79 View Post
2. Relevant equations

##\nabla f = f_z i+ f_y j+f_z k##
Shouldn't this be: [itex]\nabla f = f_x i+ f_y j+f_z k[/itex]?

bugatti79
#4
May12-12, 04:13 PM
P: 652
Calculus Identities

Quote Quote by gopher_p View Post
Any chance ## f \nabla g +g\nabla f ## and ## f \nabla f## might be conservative?
Quote Quote by sharks View Post
In my opinion, there are some missing information in your problem.

Is ##g## also a continously differentiable real valued function in R^3?


Shouldn't this be: [itex]\nabla f = f_x i+ f_y j+f_z k[/itex]?
Yes, you are correct sharks, that is a typo. It should be as you have stated.
##...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
sharks
#5
May12-12, 05:40 PM
PF Gold
sharks's Avatar
P: 836
Quote Quote by bugatti79 View Post
F is a continously differentiable vector field
Do you mean ##f## or did the question involve ##\vec F##, in which case, has any information been given about the latter?
bugatti79
#6
May13-12, 03:07 AM
P: 652
Quote Quote by sharks View Post
Do you mean ##f## or did the question involve ##\vec F##, in which case, has any information been given about the latter?
Here is the proper question asked in full. Apologies again.

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
HallsofIvy
#7
May13-12, 08:01 AM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,348
Quote Quote by bugatti79 View Post
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##
This makes no sense. You have "g" in the conclusion but not in the hypotheses and "F" in the hypotheses but not in the conclusion. What is the problem, really?

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
bugatti79
#8
May13-12, 09:03 AM
P: 652
Quote Quote by HallsofIvy View Post
This makes no sense. You have "g" in the conclusion but not in the hypotheses and "F" in the hypotheses but not in the conclusion. What is the problem, really?
The correct thread/question is post #6 and not #1. The is no 'F' involved, that was in another very similar question (#1 which I will ignore). Only f and g are involved.

THanks


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