How you can say if a line integral will be independant ot a given path

[gl0ck]

Homework Statement

Here is my problem :

so far I've solved the line integral but I don't know what is the condition that must be met in order to be independent of the path given.
I found the line integral to be: 27/28

 

[xiavatar]

The vector field $\vec F$ has to be conservative.
http://tutorial.math.lamar.edu/Classes/CalcIII/ConservativeVectorField.aspx
http://en.wikipedia.org/wiki/Conservative_vector_field

 

[ehild]

Try an other path between points (0,0,0) and (1,1,1) . What about a straight line, connecting them?

ehild

 

[Feodalherren]

If the Curl of the vector field = 0 it is conservative and hence path independent.

How would you find the curl of the vector field?

 

[LCKurtz]

If the Curl of the vector field = 0 it is conservative and hence path independent.

How would you find the curl of the vector field?

Open your text and look in the index for "curl"? Or Google it? Or look at the links given in post #2?

 

[Feodalherren]

Open your text and look in the index for "curl"? Or Google it? Or look at the links given in post #2?

I'm not the one looking for help, I was trying to give it :).

 

[LCKurtz]

I'm not the one looking for help, I was trying to give it :).

Woops! My bad. Well with all the hints, maybe the OP will sometime return to the thread.

 

[HallsofIvy]

The integral of a vector function, $\vec{F}$, is independent of the path if and only if it is "a derivative". That is, if there exist a real-valued a function, f, such that $\nabla\cdot f= \vec{F}$. That will be true for this vector function if $f_x= xy$, $f_y= yz$, and $f_z= xz$.

We can check if that is true by looking at the mixed second derivatives: $f_{xy}= x$ and $f_{yx}= z$. Those are NOT the same so this function is NOT independent of the path.