# Thread: Urgent: Review problem for Final: Path Independence and Potential Functions

1. ## Urgent: Review problem for Final: Path Independence and Potential Functions

This isn't homework but it's on our review for our final and I can't figure out the second part to save my life.

Question: Let $\displaystyle F(x,y) = (2x + y^{2} + 3x^{2}y)i + (2xy + x^{3} + 3y^{2})j$. Show that the line integral $\displaystyle \int_c F*dr$ is independent of the path, and then find the value of the integral from (0,1) to (5,3).

The * is supposed to mean dot.

I know you're supposed to have a F(x) = the i part and F(y) = the j part, take the derivative with respect to x of the i part and take the derivative with respect to y for the j part. I don't know where to go from there.

thanks

2. Originally Posted by FalconPUNCH!
This isn't homework but it's on our review for our final and I can't figure out the second part to save my life.

Question: Let $\displaystyle F(x,y) = (2x + y^{2} + 3x^{2}y)i + (2xy + x^{3} + 3y^{2})j$. Show that the line integral $\displaystyle \int_c F*dr$ is independent of the path, and then find the value of the integral from (0,1) to (5,3).

The * is supposed to mean dot.

I know you're supposed to have a F(x) = the i part and F(y) = the j part, take the derivative with respect to x of the i part and take the derivative with respect to y for the j part. I don't know where to go from there.

thanks
Find a scalar function $\displaystyle \phi$ such that $\displaystyle F = \nabla \phi$:

$\displaystyle \frac{\partial \phi}{\partial x} = 2x + y^2 + 3x^2 y$ .... (1)

$\displaystyle \frac{\partial \phi}{\partial y} = 2xy + x^3 + 3y^2$ .... (2)

Solve the above simultaneous partial differential equations for $\displaystyle \phi$.

The line integral is equal to $\displaystyle \phi(5,3) - \phi(0,1)$.
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