Substitution rule, transformation formula, change of variables

Here, I present the substitution rule that is needed in Multivariable Calculus. Often, it is just called change of variables since we switch to a better coordinate system that makes the domain of integration easier and hopefully also the function.

I explain the theorem in all details and do some sketches.

I apologise for my pronunciation. The focus is on the mathematics and not my English skills :)

(This exercise fits to lectures for students in their first and second year of study: Mathematics for physicists, Mathematics for the natural science, Mathematics for engineers, Linear algebra and so on)