partial derivative of a function defined between manifolds I know the definition of derivative of a map $\phi : M \to N$ defined between manifolds but i am willing to know that how the partial derivative of $\phi : M \to N$ will be defined? I know the definition of partial derivative if $N = \mathbb{R}$. 
Also, i am not able to see that what would be the geometric interpretation of the partial derivative of a function between manifolds? We know that in multivariable analysis partial derivative represents  just the directional derivative with respect to  a coordinate axis. 
Thanks!

There is no meaning of partial derivative of your function. But if you want to discuss the differentiability of $f$ we use the following process.
Suppose that $U$ is a neighborhood of a point $p\in M^m$ and $V$ is a neighborhood of $f(p)\in N^n$. Also let the charts on $M$ and $N$ are $(\rho,U),(\sigma,V)$ about both $p,f(p)$ respectively. Then the function $f$ is called differentiable if the function
$$\sigma^{-1}\circ f\circ \rho:\mathbb{R}^m\to\mathbb{R}^n$$ is differentiable.
