If $c:I^k\rightarrow\mathbb{R}^n$ is a singular $k$-cube.
What will $\partial c$ be?
Will it be a singular $(k-1)$-cube or a $(k+1)$-cube or something else?
Definition - A singular $k$-cube on $U$ is a smooth map $c:I^k\rightarrow U$
where $I^k=\{t\in\mathbb{R}^k\mid 0\leq t^i\leq 1\}$ denotes the closed unit $k$-cube in $\mathbb{R}^k$