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Change of variables is a basic method in mathematical solving. However, I can't use it smoothly, i.e. I don't know when to use it. (I can use it in some regular problems, but I won't have the sense of using it when coming across some new problems). Besides that, I'm not sure why we can use the change of variables,i.e.why the change of variable is reasonable.For example, in multivariable calculus, it is largely used. Can anyone give me some ideas?

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What do you mean by "variance substitution"? There may be a language barrier here: perhaps you mean "change of variables"? – Robert Israel Oct 31 '12 at 22:09
@RobertIsrael Yeah,I have corrected – 89085731 Oct 31 '12 at 22:21
up vote 8 down vote accepted

This is an excellent question, but one which is quite difficult to answer. "Changing variables" is an example of a problem solving heuristic (or "strategy") that can be very powerful when used in the right scenario.

Perhaps if you were asked to solve for $x$ in the following you might try substitution:

$$x^4 - 6x^2 + 8 = 0$$

This doesn't change the structure of the original problem (and after a bit of practice you might find a substitution of $y = x^2$ unnecessary for factoring the left-hand expression above) but it can drastically change your perspective on the problem. Indeed,

$$y^2 - 6y + 8 = 0$$

is easily seen as a quadratic equation, where the roots can be found by factoring (or, if you are feeling particularly obstinate, using the quadratic formula).

If you are interested in reading about problem solving heuristics, a good place to start is the work of George Polya and Alan Schoenfeld.

In general, it turns out to be very difficult to develop a sense of when to use which heuristic. For your particular question, this means that it is tough to explain when you should use a change of variables (or some other strategy). I suppose it would be possible to start a list of situations in which you might use a variable change, but doing so would either require too much generality to be helpful (Polya's seminal work "How to solve it" suffers from this problem) or too specific to be of a reasonable length or readable.

The best way to learn about how/when to change variables (in fact, the only way that has proven to be effective) is by solving lots and lots of problems. Heuristic use boils down to intuition, and intuition is developed as you are exposed to many different situations and have to weasel your way out of them.

I realize this answer, though true, is somewhat disappointing. So let me end with a nice problem that can be solved using substitution (in a couple different ways). Solve for $x$ in the following equation: $$(x-1)(x-2)(x-3)(x-4) = 2013$$

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