= Solution
{c}
Evaluate the <line integral> separately on the three oriented <line segments>. Their <parametrizations> and contributions are
$$
\begin{array}{c|c|c}
\mathbf r(t),\ 0\le t\le1&\mathbf F(\mathbf r(t))\cdot\mathbf r'(t)&\text{contribution}\\\hline
(t,0,0)&0&0\\
(1,t,0)&0&0\\
(1,1,t)&t&1/2
\end{array}
$$
Thus
$$
\boxed{\int_C\mathbf F\cdot d\mathbf r=\frac12.}
$$
The different value in <3/b/i> demonstrates <path dependence>. The <vector field> in this part is different from the <conservative vector field> in part (a).
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