Calculating Constants for Conservative Vector Field F

[Bennyboymalon]

We are told that the force field F=($$\mu$$z + y -x)i + (x-$$\lambda$$z)j + (z+($$\lambda$$-2)y - $$\mu$$x)k

Having already calculated in the previous parts of the question
1:the line integral (F.dx) along the straight line from (0,0,0) to (1,1,1)
2:the line integral (F.dx) along the path x(t)=(t,t,t^2) from (0,0,0) to (1,1,1)

returning the answers
1: 0
2: ($$\lambda$$ - $$\mu$$ - 1)/3

The final part of the question (which I need help on please!) asks to calculate values for $$\mu$$ and $$\lambda$$ for which F is conservative.

I know that F.dx =0 for a conservative field and that the value of the line integral is path independant. Can therefore equate the answer of part 2 to zero, thus returning;

$$\lambda$$ = 1 + $$\mu$$ and/or $$\mu$$ = $$\lambda$$ - 1

but then how is this useful?
Could use curl F = 0 but curl is a vector and so not useful in returning scalar values for the two constants.
Could calculate an expression for f (x,y,z) as gradf=F but i have tried this and it gets pretty complicated and of course don't have any numerical value to equate this to, so that subbing in expressions for the constants could help return a value for each. I'm going to keep trying it but any hints on how to solve this problem would be much appreciated!
many thanks

 

[xepma]

First, use your results to write $$\mathbf{F}$$ as a function of the parameter $$\lambda$$ alone (or only in terms of $$\mu$$, whichever you prefer. By doing this you have elimnated one variable.

The reason for this is that, as you know, this is a necessary condition for the vector field to be conservative. However, it may not be a sufficient one. Still, the elimination of one variable is a good step.

Next, you can apply one of the methods you stated - I would personally suggest evaluating the cross product: $$\nabla\times\mathbf{F}=0$$. This gives you three equations in terms of x,y,z and $$\lambda$$. These should hold for all values of x,y,z.

Now two things can happen. Either from these equations you can solve for a particular $$\lambda$$, which is ofcourse what you would like in the first place. Or you get an equations that reduces to something like: $$\lambda= a*x$$. That is, you cannot eliminate the last variable. In that case you are not dealing with a particular value for $$\lambda$$ for which the field is conservative, but a whole range - which is perfectly fine by the way.

 

[Bennyboymalon]

Thanks for your quick reply! I have calculated curl f and found it to be;

(z-x(1+$$\mu$$)+2$$\mu$$)i + (-x-z+$$\mu$$y)j + (x-z(1+2$$\mu$$)k

which is = 0 for a conservative field. So how then can I solve this vector expression to get a value for $$\mu$$. Do i have to do the modulus of curl F and go from there? I'm a little confused.

 

[xepma]

Note that your answer is a vector. So you actually have to set the curl equal to the null vector, ie. 0*i + 0*j + 0*k. So strictly speaking, it is indeed not equal to the number 0 ;).

In other words, each component of the vector needs to be zero "by itself". This gives you three equations.

 

[Bennyboymalon]

Fantastic, I thought it was probably that and have got some answers. If you wanted to know;
$$\mu$$ = 0 and $$\lambda$$= 1
which i have then re-entered into the original definiton of F, and it does indeed return that the integral of F. dx is = 0.
Thank you very much for your help! Much appreciated! :)

 

[xepma]

Your welcome :)