I would reformulate the problem in the following manner:

Given is a manifold with codimension 1 (→ 1 equation), the hypersurface $H$ and a manifold with dimension 1 ($n-1$ equations), the curve $C$.

Now a necessary condition for $C\in H$ is $$\rm d h \wedge \rm d c_1 \wedge \ldots \wedge \rm d c_{n-1} = 0,$$

where $$h(x_1, ..., x_n) = 0$$ characterizes $H$ and the equations $$c_i(x_1,..., x_{n-1})=0, \text{ for } i = 1, ..., n-1 $$ characterize $C$.

I would reformulate the problem in the following manner:

Given is a manifold with codimension 1 (→ 1 equation), the hypersurface $H$ and a manifold with dimension 1 ($n-1$ equations), the curve $C$.

Now a necessary condition for $C\in H$ is $$\rm d h \wedge \rm d c_1 \wedge \ldots \wedge \rm d c_{n-1} = 0,$$

where $$h(x_1, ..., x_n) = 0$$ characterizes $H$ and the equations $$c_i(x_1,..., x_{n})=0, \text{ for } i = 1, ..., n-1 $$ characterize $C$.