Covariant Derivative of Contravariant component

Question

I was reading about the Voss-Weyl formula for the computation of Divergence of vector in curvilinear coordinate system. From my understanding the Divergence is equivalent to covariant derivative of contravariant components of vector, so can I replace the covariant derivative of contravariant components of a vector everytime with Voss-Weyl formula?

$$\boldsymbol\nabla \cdot F=\frac{1}{\sqrt{\vert g \vert}} \partial_i\left(\sqrt{\vert g \vert}F^i\right)$$

Answer 1

You are right, it is the same divergence of a contravariant tensorfield. The V-W formula calculates it in a nontensor way without Christoffel symbols. (Pavel Grinfeld: Introduction to Tensor Analysis and the Calculus of Moving Surfaces, p. 144-145)