Answer
(a) See the explanation below.
(b) See the explanation below.
(c) See the explanation below.
Work Step by Step
a) Suppose $F$ is a scalar function defined on the curve $C$ which is parameterized by a vector function $r(t)$ in the close interval $[m,n]$ .
$\int_C F.dr=\int_Cf(r(t))ds=\int_m^nf(r(t))r'(t)dt$
b) From part (a), we have $\int_Cf(r(t))ds=\int_m^nf(r(t))r'(t)dt$. Here, since $F$ defines a force field, then the line integral represents the total work done or energy required to move an object from point $n$ to $m$.
c) As we are given: $F=\lt P,Q,R \gt $
Thus, $\int_C F \cdot dr=\int_C(Pdx+Qdy+Rdz)$
