Change of variables - Wikipedia
In mathematics, a
change of variables is a basic technique used to simplify problems in which the original
variables are replaced with
functions
of other variables. The intent is that when expressed in new variables,
the problem may become simpler, or equivalent to a better understood
problem.
Change of variables is an operation that is related with
substitution. However these are different operations, as it can be seen when considering
differentiation (
chain rule) or
integration (
integration by substitution).
A very simple example of a useful variable change can be seen in the problem of finding the roots of the sixth order polynomial:
Sixth order polynomial equations are generally impossible to solve in terms of radicals (see
Abel–Ruffini theorem). This particular equation, however, may be written
(this is a simple case of a
polynomial decomposition). Thus the equation may be simplified by defining a new variable
x3 =
u. Substituting
x by
u
3
{\displaystyle {\sqrt[{3}]{u}}}
into the polynomial gives
which is just a
quadratic equation with solutions:
The solutions in terms of the original variable are obtained by substituting
x3 back in for
u:
Then, assuming that
x is
real,
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