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What are Contravariant and Covariant Components of a Vector? Part 2

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Uploaded on May 22, 2009

Update DEC 18 2013 with link to Dual Space of Linear Functionals. The contravariant and covariant components of a vector is central concept of differential geometry. Here we discuss why they are useful and necessary to extend the concept of the length of a vector to a large class of coordinate systems. An example coordinate system having oblique axes is analyzed and explicit formulas for the contravariant and covariant components are found. We use a simple geometric approach to the problem that provides a foundation for understanding tensors and the metric tensor in higher dimensions.

Animation videos showing covariant and contravariant components of a vector Q and the associated oblique coordinate systems in a plane: http://www.youtube.com/watch?v=hu_tmB... AND the covariant and contravariant basis vectors (dual basis vectors) illustrated in: http://www.youtube.com/watch?v=NaR0Zu... AND the length of a vector as an invariant under oblique coordinate angle changes illustrated in: http://www.youtube.com/watch?v=3NZQga... also Dual Space of Linear Functionals LINK: http://www.youtube.com/watch?v=6ckM1-...

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