# Math Help - Math 264 Review

1. ## Math 264 Review

Hi there, I just wandered in. I thought I had a fairly good idea of how to do #1, but I'm not too fond of the answer I keep finding. The problem in question is at the top of my scanned sheet dealing with work and line integral.

2. Note the vector field is conservative

$f(x,y,z)=\int2y^2zdx=2xy^2z+g(y,z)$

$f_y(x,y,z)=4xyz+g_y(y,z)=4xyz$

so g can only be a function of z

$f_z(x,y,z)=2xy^2+g_z(z)=2xy^2+1 \iff g_z(z)=1 \iff g(z)=z$

so our function is

$f(x,y,z)=2xy^2z+z$

by the fundemental theorem of line integrals we can just evaluate at the end points for the value.

$r(0)=(1,0,0)$
$r(\pi)=(-1,0,\frac{\pi}{6})$

$f(-1,0,\pi/6)-f(1,0,0)=2(-1)(0)^2(\pi/6)+\pi/6-(2(1)(0)^2(0)-0)=\frac{\pi}{6}$

I hope this helps.

3. It sure did. Thanks a bunch.