# Thread: Conservative Vector Field Function

1. ## Conservative Vector Field Function

Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F= ∇f.

F(x, y, z) = 2xy i + (x^2 + 2yz) j + y^2 k

I found the vector field to be conservative but how do I find the function?

The book's answer is x^2 • y + y^2 • z + K

2. ## Re: Conservative Vector Field Function

Originally Posted by USNAVY
Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F= ∇f.
F(x, y, z) = 2xy i + (x^2 + 2yz) j + y^2 k
I found the vector field to be conservative but how do I find the function?
The book's answer is x^2 • y + y^2 • z + K
I like the notation: $F(x,y,z)=I(x,y,z)\bf{i}+J(x,y,z)\bf{j}+K(x,y,z) \bf {k}$ it makes it to keep up with which is which.
In the above, $I(x,y,z)=2xy,~J(x,y,z)=x^2+2yz,~\&~K(x,y,z)=y^2$
For conservative note: $\begin{array}{l}K_y=J_z=2y\\K_x=I_z=0\\J_x=I_y=2x \end{array}$

This will always give you the primitive:
$\displaystyle{\Theta (x,y,z) = \int_0^x {I(t,0,0)dt} + \int_0^y {J(x,t,0)dt} + \int_0^z {K(x,y,t)dt}}$

See if you get what the text has. I did.