I am reading this
https://en.wikipedia.org/wiki/Bochner_identity
What is the definition of a gradient of $f:M\rightarrow N$?
I only know that of a function $f:M\rightarrow\mathbb{R}$ is defined as
$<\nabla f,X>=X(f)=X^i\frac{\partial f}{\partial x^i}$, so $\nabla f=g^{ij}\frac{\partial f}{\partial x^j}\partial_i$. But I have no idea what it is if $f=(f^1,\dots,f^n)$ is $n$-dimensional. At first I guess it might be like a matrix, but if so what is $|\nabla f|$?