# Earth Cubed

## Vector Identities and Derivations

To derive the prognostic equations in a GCM usually requires the use of vector identities. Here is a list of some links which give vector identities:

I’ve been searching for the derivations of these identities and have found an excellent source:

Vector algebra is a powerful and needful tool for Physics but unfortunately, due to lack of mathematical skills, it becomes misleading for first undergraduate courses of science and engineering studies. Standard vector identities are usually proved using Cartesian components or geometrical arguments, accordingly. Instead, this work presents a new teaching strategy in order to derive symbolically vector identities without analytical expansions in components, either explicitly or using indicial notation. This strategy is mainly based on the correspondence between three-dimensional vectors and skew-symmetric second-rank tensors. Hence, the derivations are performed from skew tensors and dyadic products, rather than cross products. Some examples of skew-symmetric tensors in Physics are illustrated.

This will require some understanding of some concepts of tensor algebra. Here are some helpfull links of concepts that will be needed:

As a side note, the prognostic equations are typically derived from the vorticy equation.  See:

August 23, 2009 - Posted by | Math