A vector field $V$ is in P;
that is $V^c $ if we considere $x^c$, $V^b$ if we consider $x^b$.
$\nabla_b V^c$ means that we evaluate the variation of $V$ components in passing from $x^c$ to $x^b$ (and it isn't a tensor)
The "rate of change" is expressed by the Jacobian matrix in P.
We know how to find derivatives of functions $F:R^2 \rightarrow R^2$ Wikipedia - Jacobian:
a nonlinear map sends a small square to a distorted parallelogram.
The Jacobian at a point gives the best linear approximation of the distorted parallelogram near that point,
and the Jacobian determinant gives the ratio of the area of the approximating parallelogram to that of the original square. $$ $$ Covariant derivative is a tensor; in a not flat ST, we have to consider another system, $ x ^q $,
and $ \quad \Gamma_b{}^c{}_q V^q \quad$ describes the correction due to the ST curvature.
$$ $$ Covariant derivative: $$\nabla_b V_c \equiv \partial_b V_c -\Gamma_b{}^q{}_c V_q$$
Is $\Gamma_b{}^c{}_q = \frac{1}{2}g^{cd}\Big(g_{bd,q}+g_{dq,b} -g_{bq,d} \Big)$ ?
how do you interpret the index position of $ \Gamma $ (q is to the right of $ b $, and to the left of $ c $) ?
I would like to know if, similarly, $\quad \nabla_b V^c \equiv \partial_b V^c +\Gamma_b{}^c{}_q V^q \quad $
A vector is a tensor, it has a physical meaning in a contra-variant form;
what does it represent in covariant form ?
