What is Jacobian Mapping?

During the assembly of the global stiffness matrix (K), local stiffnesses are numerically integrated in the parent coordinate (the polynomial is a function of s). This is better illustrated by considering the following element:

which sits in a normalised system that varies between -1 and +1. This requires to be mapped to the actual element coordinate (change to a function of x):

This mapping requires the use of Jacobian matrix (J), which is commonly used for coordinate transformation in mathematics.