In the book I am reading, the Boundary of a $(k+1)-$cell $\varphi$ is defined to be the $k-$chain
$$\partial\varphi=\sum_{j=1}^{k+1}(-1)^{j+1}(\varphi\circ \iota^{j,1}-\varphi\circ \iota^{j,0}) $$
Where $$ \iota^{j,0}:(u_1,\ldots,u_k)\mapsto(u_1,\ldots,u_{j-1},0,u_j,\ldots,u_k)$$ $$ \iota^{j,1}:(u_1,\ldots,u_k)\mapsto(u_1,\ldots,u_{j-1},1,u_j,\ldots,u_k)$$ I am struggling to reconcile this with the intuitive definition of the boundary of a region. Also, I am uncertain as to why there is an alternating negative/positive sign attached to this sum.
Any help would be greatly appreciated.
NOTE: Here a cell denotes a smooth map from $I^k$ to $\mathbb{R}^n$, rather than the image of the smooth map. Apparently it can mean one or the other in different cases.