How Do You Calculate the Potential Function for a Vector Field?

[Chandasouk]

Finding potential function??

F=<2xy+5, x²-4z, -4y>

How do you get the potential function of this?

I perform it in this manner

<Fx, Fy, Fz> = <2xy+5, x²-4z, -4y>

So, Fx = 2xy+5 ; Fy = x²-4z ; Fz = -4y

I take the integral of Fx to obtain x²y+ 5x + h(y,z) = f which is the variable i am using to represent potential function.

I take the partial derivative with respect to y of f and obtain

Fy = x² + h_y(y,z). The set the two different Fy equations equal to each other

x²-4z = x² + h_y(y,z)

-4z = h_y(y,z)

How do I solve from here? The -4z left over throws me off.

 

[ehild]

-4z = h_y(y,z)

How do I solve from here? The -4z left over throws me off.

You know the derivative of h with respect to y. It does not depend on y, -4z can be considered a constant. Integrate.

ehild

 

[Chandasouk]

Integrate with respect to y? The final answer is x^2 + 5x - 4yz

If you integrate -4z = hy(y,z) with respect to y, that leaves me with -4yz + g(z) doesn't it? What happened to that g(z)?

 

[ehild]

Derive with respect to z and compare to Fz.

ehild