Is It Possible to Prove the Conservativity of These Force Fields?

[Hoppa]

hello I am having a few troubles on these two force field problems, determining whether that are conservative or not.

F = (x, y, z) / (x^2 + y^2 + z^2)^3/2

and

F = (x, y, z) / (x^3 + y^3 + z^3)

i know that when the force is independent of the path then the force is said to be conservative, but how to i show that.
also i know i got to use the equation
W = Integral (t1 - t0) F mulitplied by v dt
but how do i implement this?

 

[James R]

Are you familiar with the curl of a vector field?

F is conservative if $\nabla \times F = 0$.

 

[Theelectricchild]

you can also check your work--- that is, when a vector field is curl-free, it will do no work on, for example, a particle along a closed path.

 

[Hoppa]

no i not familiar with the curl of a vector field? should i be to be able to do these questions?

 

[HallsofIvy]

hello I am having a few troubles on these two force field problems, determining whether that are conservative or not.

F = (x, y, z) / (x^2 + y^2 + z^2)^3/2

and

F = (x, y, z) / (x^3 + y^3 + z^3)

i know that when the force is independent of the path then the force is said to be conservative, but how to i show that.
also i know i got to use the equation
W = Integral (t1 - t0) F mulitplied by v dt
but how do i implement this?

A force field is conservative if and only if there exist some "potential" from which the force can be derived. That is, mathematically, if there exist some scalar function $\phi$ such that $F= \nabla \phi$.
That is, the x coordinate of F is the derivative of [itex]\phi[/tex] with respect to x, the y coordinate the derivative with respect to y, the z coordinate the derivative with respect to z.

It isn't necessary to FIND $phi$ to determine if F is conservative- use the fact that the mixed second derivatives are independent of order (as long as they are continuous- which is true here). That, if F(x,y,z) is given by the vector f(x,y,z)i+ g(x,y,z)j+ h(x,y,z)k then F is conservative if and only if
$f_y= g_x, f_z= h_x, g_z= h_y$ (subscripts indicate partial derivatives).

 

[Hoppa]

A force field is conservative if and only if there exist some "potential" from which the force can be derived. That is, mathematically, if there exist some scalar function $\phi$ such that $F= \nabla \phi$.
That is, the x coordinate of F is the derivative of [itex]\phi[/tex] with respect to x, the y coordinate the derivative with respect to y, the z coordinate the derivative with respect to z.

It isn't necessary to FIND $phi$ to determine if F is conservative- use the fact that the mixed second derivatives are independent of order (as long as they are continuous- which is true here). That, if F(x,y,z) is given by the vector f(x,y,z)i+ g(x,y,z)j+ h(x,y,z)k then F is conservative if and only if
$f_y= g_x, f_z= h_x, g_z= h_y$ (subscripts indicate partial derivatives).

ok what does that mean?

 

[whozum]

It means for vector field f(x,y,z), it is conservative if and only if the partial derivatives mentioned in his post are equal. If you have f(x,y,z) you can take these partial dreivatives and compare.

 

[Hoppa]

ok how do i show whether it is conservative or not? like what is the working that i need?

 

[whozum]

Can you break your vector field into its components?

Is it F(x,y,z) = (x^2i + y^2j + z^2k)^(3/2)?

If so, I don't think it can get simpler than what HallsofIvy said:

That is, if F(x,y,z) is given by the vector f(x,y,z)i+ g(x,y,z)j+ h(x,y,z)k then F is conservative if and only if
(subscripts indicate partial derivatives).

 

[Theelectricchild]

Understanding the basics of divergence and curl are essential to know the meanings of the math that you are doing--- if your instructor has not explained the curl of a vector field to you, it baffles me as to why you are required to solve the aforementioned problem.

 

[whozum]

Understanding the basics of divergence and curl are essential to know the meanings of the math that you are doing--- if your instructor has not explained the curl of a vector field to you, it baffles me as to why you are required to solve the aforementioned problem.

We learned curl and divergence way later than what he's doing here. curl and divergence were the last section in vector calculus here.

 

[Hippo]

Really? That makes no sense whatsoever. :yuck:

(Make no mistake, though; I believe you.)

 

[whozum]

Surely youve done this kind of problem in class, without using curl of divergence. Do you have ANY kind of idea on what to do?

 

[James R]

Of course, you CAN show that a force is NOT conservative without using curl. All you need to do is to choose two different paths in space and show that the work done by the force along each path is different.

 

[Hippo]

Surely youve done this kind of problem in class, without using curl of divergence. Do you have ANY kind of idea on what to do?

ANY kind of idea? Well, yeah.


I would evaluate $$\nabla \times \vec{F}$$.

That's a simple and straightforward way.

Look, I understand the definition of a conservative field, and, as it demands a "potential" function, it implies that the curl of a conservative field will always be zero.

I think one or two have suggested evaluating $$\oint_C \vec{F} \bullet ds$$ which is fine and dandy.

However, line integration over a closed curve is really a silly way of doing it; the Curl Theorem implies that such a thing will vanish so long as the curl(F) is zero, and the integration itself is guarunteed to be more difficult than evaluating $$\nabla \times \vec{F}$$.


I was just commenting on how strange the subject order you mentioned seemed to me, nothing more.

All you need to do is to choose two different paths in space and show that the work done by the force along each path is different.

Can you clarify that, James? Or explain further?

 

[jdavel]

HallsofIvy,

But fy = gx, fz = hx, gz = hy are exactly the conditions that the z, y and x components (respectively) of curlF are all zero. Right?

Hippo, calculating a closed line integral can disprove that F is conservative if the integral isn't zero. But if it is zero, you haven't proven that F is conservative. That requires that EVERY closed loop line integral be zero. You either need to learn HallsofIvy's conditions or learn what the curlF is.

 

[whozum]

ANY kind of idea? Well, yeah.


I would evaluate $$\nabla \times \vec{F}$$.

That's a simple and straightforward way.

Look, I understand the definition of a conservative field, and, as it demands a "potential" function, it implies that the curl of a conservative field will always be zero.

I think one or two have suggested evaluating $$\oint_C \vec{F} \bullet ds$$ which is fine and dandy.

However, line integration over a closed curve is really a silly way of doing it; the Curl Theorem implies that such a thing will vanish so long as the curl(F) is zero, and the integration itself is guarunteed to be more difficult than evaluating $$\nabla \times \vec{F}$$.


I was just commenting on how strange the subject order you mentioned seemed to me, nothing more.



Can you clarify that, James? Or explain further?

He's telling you to evaluate the line integral for two different lines.
If you know how to solve this, then what's the problem here?

edit: confused your name with the OP.

 

[James R]

Hippo:

Are you the same person as Hoppa?

I'm confused.

 

[Theelectricchild]

how could he be the same person?

 

[James R]

I assume it is possible for one person to register here under 2 different names. Isn't it?

 

[whozum]

Both people talk in the first person about the same problem, as if they have to solve it, yet one of them defines the curl function as a solution and the other seemingly has never heard of curl.

 

[Hippo]

No, I'm not Hoppa.

We learned curl and divergence way later than what he's doing here.

Whozum, you wrote that.

Really? That makes no sense whatsoever. :yuck:

I commented on your remark.

Surely youve done this kind of problem in class, without using curl of divergence. Do you have ANY kind of idea on what to do?

I assumed the above was a response to my comment and felt rather disgruntled at the way it was phrased.
That's why I gave my explanation of why I'd use curl(F) for a solution, in preference to any other method.

I'm sorry for the misunderstanding, whozum.

He's telling you to evaluate the line integral for two different lines.

"Two different lines"?

I mean to say, that's a very vague way to put it.

If he was referring to this condition for a conservative field, $$\vec{F}$$,

$$\int_{C_1} \vec{F} \bullet ds = \int_{C_2} \vec{F} \bullet ds$$

where the curves $${C_1}$$ and $${C_2}$$ have the same endpoints, then I understand completely.

However, I couldn't tell if he meant that.

 

[Theelectricchild]

I assume it is possible for one person to register here under 2 different names. Isn't it?

no, that is violation of the terms of service.

 

[whozum]

It wasnt in response to you, I'm sorry, and your names are so similar I thought the OP was talking.

The "two different lines" was in reference to

"All you need to do is to choose two different paths in space and show that the work done by the force along each path is different."
by James R

taking 'paths' as 'lines'. It lost some of its meaning in the translation I guess.