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

1. May 9, 2014

gl0ck

1. The problem statement, all variables and given/known data

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 independant of the path given.
I found the line integral to be: 27/28

Last edited: May 9, 2014
2. May 9, 2014

3. May 9, 2014

ehild

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

ehild

4. May 9, 2014

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?

5. May 9, 2014

LCKurtz

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

6. May 9, 2014

Feodalherren

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

7. May 9, 2014

LCKurtz

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.