I have this question:
Calculate the integral $∫F · dr$ when $F = (−x^2y, xy^2, 0)$ and $C$ is a square in the $(x, y)$−plane with vertices at $(0, 0), (l, 0), (l, l), (0, l)$ which is oriented anticlockwise.
I know the formulae for the line integrals of scalar and vector fields, however am a bit confused with how to apply them- I know you need a path, which connects points along a curve, but I cannot see what the path is in the above question.
The path $\vec c$ is the combination of the 4 line segments of the square, namely:
$\vec c_1$ is the line segment from $(0,0)$ to $(l,0)$
$\vec c_2$ is the line segment from $(l,0)$ to $(l,l)$
$\vec c_3$ is the line segment from $(l,l)$ to $(0,l)$
$\vec c_4$ is the line segment from $(0,l)$ to $(0,0)$.
And the total path $\vec c$ is $\vec c = \vec c_1 + \vec c_2 + \vec c_3 + \vec c_4$
But you can avoid parametrising each of the lines and doing 4 line integrals by using Green's theorem if you know what that is...