I am thinking about the intrinsic meaning (what this equation really means) about this equation.
Suppose $\mathcal M$ is a smooth manifold embedded in $\mathcal R^d$, then for any $x \in \mathcal M$,
dim($\mathcal M$) = $\sum_{i=1}^d ||\nabla_\mathcal M \alpha_i (x)||$,
where $\nabla_\mathcal M$ is the gradient on the manifold and $\alpha_i (x)$ is the coordinate function, i.e $\alpha_i (x)=x_i$, where $x=(x_1,...,x_d)$
I know how to prove this but what does it really mean behind this equation? Can someone help me? Thanks!