Scalar potential and line integral of a vector field

[alecst]

Homework Statement

Homework Equations

Given above.

The Attempt at a Solution

I attempted this problem first without looking at the hint.

I've defined F(r) as (B+A)/2 + t(B-A)/2, with dr as (B-A)/2 dt . Thus F(r)dr = ((B+A)/2)*((B-A)/2)+((B-A)/2)^2 dt

When I integrate this from -1 to 1 I get 1/2*(B^2-A^2).

When I then looked at the hint, I saw it mentioned another (B^2+A^2)/2 term and another "c," neither of which I have, and my integrand has no "tau" squared element either. Is there a point where I went wrong here?

 

[LCKurtz]

I'm guessing that you don't get to define F(r) but instead have to use the one given to you in problem 4.01, whatever that is.

 

[alecst]

The idea is to used the derived formula to solve the next problem, which is find a scalar potential function phi(r) such that the line integral F(B,A) (as in 4.02) = phi(B)-phi(A). So it's clear I need to solve this in terms of B and A.