How do I find the potential function for this vector field?

[Chandasouk]

I need to find the potential function of this Vector Field

F= <2xy+5, x²-4z, -4y>

I already checked that a potential function does exist.

I made F1 = 2xy+5, F2= x²-4z, and F3=-4y

I first integrated F1 with respect to x

$\int (2xy+5)dx$

and got x²y+5x+ g(y,z)

so our potential function,f, is currently x²y+5x+ g(y,z)

I then take the partial derivative of f with respect to y

df/dy(x²y+5x+ g(y,z)) = x²+ dg/dy

Now, I set this equal to F2 which yields

x² -4z = x²+ dg/dy

-4z = dg/dy

Then to obtain g(y,z) I do

$\int (-4z)dy = -4zy+h(z)$

Can anyone show me how to complete the problem from here? I always get close to what the potential function should be but mess up at the end. I need to know this for my final coming up.

 

[phyzguy]

Just keep going. Now you have a new expression for f. Take $$\frac{\partial f}{\partial z}$$ and set it equal to Fz, just like you did in the 2nd step.