The Weitzenböck formula for the divgrad operator

The paper concerns symmetric tensors and consists of interesting formulas having also consequences in geometry. First, the differential operators of gradient and of divergence in the bundles of symmetric tensors and symmetric tensors with values in the tangent bundle, respectively, are investigated, leading to a new, simple proof of the Weitzenböck formula. In the second part of this paper the differential operators and symmetric forms on $\mathbb R^n$ are investigated as an application of previous considerations.