In our course on several variable calculus,the following notion was defined:
Vector Field along a parametrization:
Definition:
Suppose $M$ is a $k$-manifold in $\mathbb R^n$.Let $q$ be a point on the manifold and define local parametrization $C^\infty(U,V)\ni\psi(=\psi(u_1,u_2,...,u_k)):U\to V$ where $U$ is an open set in $\mathbb R^k$ containing $p$ such that $\psi(p)=q\in V$ which is open in $M$.We let the coordinates in $U$ be $(u_1,...,u_k)$ and the coordinates in $\mathbb R^n$ be $(y_1,...,y_n)$.Define $X_i(p)=D\psi(p)(\frac{\partial}{\partial u_i}|_p) $ where $X_i:U\to T_q(M)\subset T_q(\mathbb R^n)$.Then $X_i(p)=\sum\limits_{j=1}^n \frac{\partial \psi_j(p)}{\partial u_i}.\frac{\partial}{\partial y_j}|_q$ Where the coefficient functions are smooth on $U$.These $\{X_1,...,X_n\}$ are called the coordinate vector fields along the parametrization $\psi$.
These vectors $X_i(p)$ form a basis of $T_q(M)$ but they are not orthogonal.So my question is what is the interpretation of this vector field.What is basically going on?In short,I want to get a feel of what it means.