parallel vector field potential function

[jamesgibby]

If F is a conservative vector field such that every vector in F is parallel to y = x. What can we deduce about the potential function f(x,y).

I believe this means the potential function then has the form f(x,y) = c(x + y) for some c. Could someone help with proper proof?

 

[LCKurtz]

A vector parallel to [MATH]y=x[/MATH] is [MATH]\langle 1, 1\rangle[/MATH]. For each [MATH]x[/MATH] and [MATH]y=x[/MATH] the vector may have a different length and direction, depending on [MATH]x[/MATH]. So we might write the vector field as [MATH]\vec F(x,y) = \langle h(x), h(x)\rangle[/MATH], where [MATH]h(x)[/MATH] is unknown. Start with that. You are given the field is conservative. What does that tell you? Work with that.

 

[jamesgibby]

Thanks for your reply. Yeah i kind of tried that way too but never really made anything out of it.

 

[LCKurtz]

If you want help you need to show me what you are doing. For a conservative field [MATH]\vec F(x,y) = \langle P(x,y),Q(x,y)\rangle[/MATH] the condition you want is [MATH]Q_x=P_y[/MATH]. Show me what happens when you try it on this problem.