We are given \(\displaystyle \L
F(x,y) = \left( {xy^2 + 2x} \right)i + \left( {x^2 y + y + 1} \right)j\)
and \(\displaystyle \L
C:\;r(t) = (t^2 - 2)i + \left[ {\left( {t - 1} \right) + t\left( {t - 2} \right)\cos (t)} \right]j\)
to find \(\displaystyle \L
\int_C {F(x,y) \cdot dr} ,\quad 0 \le t \le 2.\) .
We can make it hard or easy. The contour is from (−2, −1) to (2,1)
It would nice, easy, if F(x,y) were conservative. Is it?
Conservative functions are path-wise independent.
Can you show that if \(\displaystyle \L
\phi (x,y) = \frac{1}{2}x^2 y^2 + x^2 + \frac{1}{2}y^2 + y\) then \(\displaystyle \L
\nabla \phi = F\) ?
Is the integral equal \(\displaystyle \L
\phi (2,1) - \phi ( - 2, - 1)\) ?
