Suppose we have have a co-ordinate mapping $\phi: M \to \mathbb{R}^n$ for a smooth Riemannian manifold $M$. Then the differential (pushforward) of this mapping $d\phi : TM \to \mathbb{R}^n\times \mathbb{R}^n$, where $TM$ is the tangent bundle, is a smooth linear map. Consider the mapping $$\sigma : M \to \mathcal{L}(T_xM,\mathbb{R}^n)\text{ given by } x\to d\phi(x),$$ where $\mathcal{L}(T_xM,\mathbb{R}^n)$ is the space of linear operators between $(T_xM$ and $\mathbb{R}^n)$. Note that the operator norm $$\| d \phi(x)\| \leq C_{x}<\infty$$ at each point $x$ (being a linear map between finite dimensional spaces). Can we claim that if $K$ is a compact set (in the usual topology for the manifold), then $$\sup_{x \in K}\|d \phi(x)\| \leq C $$ for some constant $C$.
Thanks in advanc and any help would be appreciated