Irregular and curved elements - isoparametric mapping

In the figure operation (M) represents a one-to-one transformation between the local and global coordinate systems. This in general can be shown as 

The one-to-one transformation between the global and local coordinate systems can be established using a variety of techniques (Zienkiewicz and Morgan, 

The most general method i.s a form of parametric mapping in which the transformation functions, and in Equation (2.26), are polynomials 

In general, elements with curved sides can only be generated using quadratic or higher-order master elements. 

Isoparametric transformation functions between a global coordinate system and local coordinates are, in general, written as 

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