Change of variables - Wikipedia

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:

${\displaystyle x^{6}-9x^{3}+8=0.\,}$

Sixth order polynomial equations are generally impossible to solve in terms of radicals (see Abel–Ruffini theorem). This particular equation, however, may be written

${\displaystyle (x^{3})^{2}-9(x^{3})+8=0}$

(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

${\displaystyle u^{2}-9u+8=0,}$

which is just a quadratic equation with solutions:

${\displaystyle u=1\quad {\text{and}}\quad u=8.}$

The solutions in terms of the original variable are obtained by substituting x³ back in for u:

${\displaystyle x^{3}=1\quad {\text{and}}\quad x^{3}=8.}$

Then, assuming that x is real,

${\displaystyle x=(1)^{1/3}=1\quad {\text{and}}\quad x=(8)^{1/3}=2.}$

Contents

• 1 Simple example
• 2 Formal introduction
• 3 Other examples
  
  • 3.1 Coordinate transformation
  • 3.2 Differentiation
  • 3.3 Integration
  • 3.4 Differential equations
  • 3.5 Scaling and shifting
  • 3.6 Momentum vs. velocity
  • 3.7 Lagrangian mechanics
• 4 See also

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