00:01
For this problem, we are asked to find the work done by the force field f of xy equals 2xi plus yj on a particle moving along the path counterclockwise around the triangle with vertices 0, 0, and 1 -1.
00:16
So the first step here is to parameterize our curve.
00:21
So we'd have our starting point 0, then we go to the point 10, then we go to the point 1 -0, then we go to the point 1 -1, and then we return to the start.
00:30
So i'll define this piecewise as c1, c2, and c3.
00:38
Then we can parameterize each one of those line segments.
00:43
I'll note that for each one of these, i'll be parameterizing for t between 0 and 1.
00:49
Now the first segment, i can write just as r1 of t, is going to equal t0.
00:56
For the second, we have r2 of t will equal 1 minus t, t and for the third we have r3 of t will equal zero one minus t then we'd have that r1 prime of t it's going to be one zero d t r2 prime of t will equal negative one one d t and r3 prime of t will be zero negative one d t and r3 prime of t will be we can then write that the line integral, or the were, will be equal to the line integral of f .dr across our curve or across our triangle, or in this case it would be the integral from 0 to 1 of...