# How to test if a vector field is conservative for a F(x,y,z)

• Apr 9th 2010, 09:58 AM
wizrd54
How to test if a vector field is conservative for a F(x,y,z)
How do you determine if a vector field is conservative for a fuction of x,y, and z?

For F(x,y) I know you can determine this if Py= Qx
• Apr 9th 2010, 10:03 AM
dwsmith
$\displaystyle (\frac{\partial P}{\partial y}-\frac{\partial N}{\partial z})=0$

$\displaystyle (\frac{\partial M}{\partial z}-\frac{\partial P}{\partial x})=0$

$\displaystyle (\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y})=0$
• Apr 9th 2010, 10:05 AM
wizrd54
Thanks.
• Apr 9th 2010, 10:06 AM
dwsmith
I edited my original post to make it more clear.
• Apr 9th 2010, 10:07 AM
AllanCuz
Quote:

Originally Posted by wizrd54
How do you determine if a vector field is conservative for a fuction of x,y, and z?

For F(x,y) I know you can determine this if Py= Qx

It is conservative if we can find the conditions bellow to be true:

dF1/dy = dF2/dx : dF1/dz = dF3/dx : dF2/dz = dF3/dy

Remember that there is a vector function such that

F(x,y,z) = F1(x,y,z)i + F2(x,y,z)j + F3(x,y,z)K

Fairly simple. If you want to make a non conservative field conservative, simply add in a constant infront of your F1/F2/F3, and find the value that satisfies the above equation.