Let $M$ be a $k$-dimensional manifold in $\mathbb{R}^n$. On page 122 of 'Calculus on Manifolds', Spivak defines the integral of a $p$-form $\omega$ on a singular $p$-chain on $M$,$c:I^{p}\to \mathbb{R}^n$, (At this point I assume that by 'on $M$', he simply means that $c(I^p)\subset M$), simply by the formula $$\int_c \omega = \int_{I^p} c^*\omega.$$
I have a small question about this definition: $\omega$ associates to any $x\in M$ an element of $\Lambda^p(M_x)$ (where $M_x$ is the tangent space defined earlier in the book.) Hence for the pullback $c^*\omega$, to be well defined, we need, for any $y\in I^p$, that the pushforward of a tangent vector $v \in T_y\mathbb{R}^k$, $c_* v$, belong to $M_{c(y)} \subset T_{c(y)}\mathbb{R}^n$. Must this necessarily happen by virtue of the definition or are there conditions missing?
Edit: It seems this follows quite naturally. Let $t\in I^p$, and suppose that $c(t)\in M - \partial M$ (for simplicity). Choose a coordinate system $f$ around $c(t)$, and then just consider $f_* (f^{-1} \circ c)_*$. Is this correct? Still have to go through the details.