Coordinate Transformations
摘要
We have learned two ways in \({\mathbb C} = {\mathbb R}^2\) to uniquely represent each element \(z \neq 0\) : \(z=(a,b)\) with the Cartesian coordinates a and b or \(z= (r,\varphi )\) with the polar coordinates r and \(\varphi \) . Behind this representation of elements with respect to different coordinate systems lies a coordinate transformation \((r,\varphi ) \to (a,b)\) . In \({\mathbb R}^3\) there are several such transformations of particular interest, especially cylindrical and spherical coordinates play a fundamental role in multidimensional engineering analysis, as many problems of engineering mathematics can be described and solved much more easily in specific coordinates.