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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.