Motivation: In the theory of harmonic maps between manifold, we often see the characterization $$ \Delta_g u = -g^{ij}A(u)\left(\frac{\partial u}{\partial x^i},\frac{\partial u}{\partial x^j}\right) $$ when we are working in the extrinsic viewpoint.

Question: Let $M,N$ be two Riemannian manifolds, $u:M\to N$ be a smooth map. Let $A$ denotes the second fundamental form of $N$, viewing as an embedded submanifold of $\Bbb R^k$. How is the expression $$ A(u)\left(\frac{\partial u}{\partial x^i},\frac{\partial u}{\partial x^j}\right) $$ rigorously defined?

First of all, $u$ need not be injective, so there can be a point $y\in N$ such that $u(x_1)=u(x_2)=y$. Doesn't this means that $\frac{\partial u}{\partial x^i}(y)$ is really not a vector field on $N$?

I think this non-uniqueness can be fixed for $x$ such that $du(x)$ is of full rank, so that $u$ is locally an embedding there. When $du(x)$ degenerates, however, I don't know how to make sense of it.