Change of variables is an operation that is related to substitution.
A very simple example of a useful variable change can be seen in the problem of finding the roots of the sixth-degree polynomial: Sixth-degree 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).
(Source: 1991 AIME) Solving this normally is not very difficult, but it may get a little tedious.
times continuously differentiable, bijective map from
times continuously differentiable inverse from
Some systems can be more easily solved when switching to polar coordinates.
Consider for example the equation This may be a potential energy function for some physical problem.
If one does not immediately see a solution, one might try the substitution Note that if
Then, replacing all occurrences of the original variables by the new expressions prescribed by
The chain rule is used to simplify complicated differentiation.
Then: Difficult integrals may often be evaluated by changing variables; this is enabled by the substitution rule and is analogous to the use of the chain rule above.
Difficult integrals may also be solved by simplifying the integral using a change of variables given by the corresponding Jacobian matrix and determinant.
[1] Using the Jacobian determinant and the corresponding change of variable that it gives is the basis of coordinate systems such as polar, cylindrical, and spherical coordinate systems.
The following theorem allows us to relate integrals with respect to Lebesgue measure to an equivalent integral with respect to the pullback measure under a parameterization G.[2] The proof is due to approximations of the Jordan content.
As a corollary of this theorem, we may compute the Radon–Nikodym derivatives of both the pullback and pushforward measures of
The change of variables formula for pullback measures is
The change of variables formula for pushforward measures is
As a corollary of the change of variables formula for Lebesgue measure, we have that From which we may obtain Variable changes for differentiation and integration are taught in elementary calculus and the steps are rarely carried out in full.
The very broad use of variable changes is apparent when considering differential equations, where the independent variables may be changed using the chain rule or the dependent variables are changed resulting in some differentiation to be carried out.
Exotic changes, such as the mingling of dependent and independent variables in point and contact transformations, can be very complicated but allow much freedom.
Probably the simplest change is the scaling and shifting of variables, that is replacing them with new variables that are "stretched" and "moved" by constant amounts.
This is very common in practical applications to get physical parameters out of problems.
For an nth order derivative, the change simply results in where This may be shown readily through the chain rule and linearity of differentiation.
This change is very common in practical applications to get physical parameters out of problems, for example, the boundary value problem describes parallel fluid flow between flat solid walls separated by a distance δ; μ is the viscosity and
It simplifies analysis both by reducing the number of parameters and by simply making the problem neater.
Proper scaling may normalize variables, that is make them have a sensible unitless range such as 0 to 1.
Finally, if a problem mandates numeric solution, the fewer the parameters the fewer the number of computations.
, Newton's equations of motion are Lagrange examined how these equations of motion change under an arbitrary substitution of variables
In fact, when the substitution is chosen well (exploiting for example symmetries and constraints of the system) these equations are much easier to solve than Newton's equations in Cartesian coordinates.