I know the definition of the partial derivative of a smooth function on a manifold $M$ is given by $$\partial_i f:= \left.\frac{\partial}{\partial{x^i}}\right\vert_{p}(f)=\left.\frac{\partial}{\partial{u^i}}\right\vert_{\mu(p)}(g)$$ where $\mu= (x^1,\cdots,x^d)$ is the coordinate system at a point $m\in M$, $g=f\circ\mu^{-1}$ is the coordinate expression for $f$, and $u^i$ are the Cartesian coordinates on $\mathbb R^d$. Notice the difference: $f$ is differentiated with respect to $x^i$ and $g$ with respect to $u^i$. By definition $x^i=u^i\circ\mu$. Furthermore, $\frac{\partial}{\partial{x^i}}$ is a vector field in $U$, where the chart is defined. But my question is: How is the second derivate $\partial^{2}_i$ defined? What does it really mean and where does this object live?
Excuse me if the question is very simple, but I don't know how to answer these questions. Any hint will be appreciated!!!