Circulation of a vector field, why the definition given in Feynman Lectures on Physics is true? 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 $C_t$?
here

$$\oint_{\Gamma} C\cdot\mathrm{d}s\stackrel{?}{=}\oint_{\Gamma} C_t\mathrm{d}s$$


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 $C_{t}$".
