In the third chapter of Electormagnetism, R.P.Feynman teach the basics of vectoriel integral calculus and I'm stuck in the formula $(3.30)$ give as a definition, but I don't understand why if it works for $C$ it must for $C_t$?
$$\oint_{\Gamma} C\cdot\mathrm{d}s\stackrel{?}{=}\oint_{\Gamma} C_t\mathrm{d}s$$
Physically, the integral of a vector field $C$ around a closed curve $\Gamma$ represents the work done by $C$ on a point particle that traces $\Gamma$. The work depends only on the tangential component of the field along the curve, i.e., on $C_{t}$; that is, the total work is equal to the work done by the tangential component.
In other words, at each point of $\Gamma$ decompose $C$ into tangential and normal components, say $C = C_{t} + C_{n}$. The work done in moving a point particle an infinitesimal displacement along $\Gamma$ is "all done by $C_{t}$".