Earth Cubed

Distributed Climate Science and Computing

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:

http://en.wikipedia.org/wiki/List_of_vector_identities
http://en.wikipedia.org/wiki/Product_rule
http://en.wikipedia.org/wiki/Vector_calculus_identities
http://mathworld.wolfram.com/VectorDerivative.html
http://planetphysics.org/encyclopedia/VectorIdentities.html
http://geo.phys.spbu.ru/~runov/SPIntro/SPIntro_15.pdf

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.

http://www.citeulike.org/user/pak/article/4524046
http://arxiv.org/PS_cache/arxiv/pdf/0904/0904.1814v1.pdf

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

http://www.foamcfd.org/Nabla/guides/ProgrammersGuidese3.html
http://en.wikipedia.org/wiki/Levi-Civita_symbol
http://en.wikipedia.org/wiki/Dyadics
http://en.wikipedia.org/wiki/Pseudovector

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

http://en.wikipedia.org/wiki/Vorticity_equation
http://en.wikipedia.org/wiki/Barotropic_vorticity_equation

Advertisements

August 23, 2009 - Posted by | Math

2 Comments »

  1. […] Read the original post: Vector Identities and Derivations « Earth Cubed […]

    Pingback by Math World | Vector Identities and Derivations « Earth Cubed | August 23, 2009 | Reply

  2. […] using the identity (see the thread: Vector Identities and Derivations) […]

    Pingback by Deriving The Vorticy Equation « Earth Cubed | August 29, 2009 | Reply


Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: