After reviewing change of variables, I realized that every text I read provided the change of variable equations in the problem descriptions. From what I understand, the main use of changing variables before integrating is to make a difficult integral into an easier integral. Problems given in textbooks or online give you a set of transformation equations that happen to nicely transform difficult integrals into integrals of relatively simple functions over relatively simple regions. If I were given a difficult looking integral "in the wild", I would have little idea about how come up with transformation equations that effectively simplify the calculations.

Suppose I am given a double or triple integral of a particular two or three variable function over a known region. For simplicity, suppose that the function and region are real-valued. How can I come up with a set of transformation equations that transform the given region into a specific region?

As examples, how could I transform any 2D region into a circle or a rectangle. In 3D, how could I transform any 3D region into a sphere or rectangular prism. These examples assume that a two or three variable function and 2D or 3D region are given.

I am guessing that simplifying the transformed region does not necessarily simplify the transformed function for integration. So, additionally, I would appreciate advice on intuiting what region to transform to for a given function and region.

  • $\begingroup$ In actual practice, it will rarely be the case that a change of variables will make the region of integration any nicer. These kinds of problems are mostly cooked up to teach the method. Just like "most" expressions you write down do not have an elementary antiderivative, and yet we make people churn out integral after integral. $\endgroup$ – Steven Gubkin Jul 13 '15 at 22:17
  • $\begingroup$ So, is the change of variables only used when you already know the transformation equations between coordinate systems? What other applications are known? Couldn't it be possible to transform an integral in a way that makes a non-integrable function integrable? If so, is there a systematic way of doing it (at least for certain types of functions)? If not, how are you so sure? $\endgroup$ – SamWeiss Jul 13 '15 at 23:15
  • $\begingroup$ Even in one variable, there are some elementary functions whose antiderivatives are not elementary. In other words, you can "write down a formula" for the function, but not its antiderivative, no matter what you do (no crazy change of variables or integration by parts or whatever can ever work). See en.wikipedia.org/wiki/… for example. Change of variables (in one and several variables) is still a valuable tool, and something you just should understand. $\endgroup$ – Steven Gubkin Jul 14 '15 at 2:02
  • $\begingroup$ Different people will have different reasons for caring. If you are an engineer, you probably only ever need to numerically approximate your integral. Even in this case, clever variable subs can be helpful, for example to eliminate a pole, which would be numerically unstable. If you are an mathematician you might use variable substitution every day in a more theoretical context. For instance, there is a whole field of study called differential geometry, and the basic objects of study use change of variables in there very definition (to "patch coordinates"). $\endgroup$ – Steven Gubkin Jul 14 '15 at 2:05

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