Change of variables

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 to substitution. However these are different operations, as 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-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). Thus the equation may be simplified by defining a new variable formula_3. Substituting "x" by formula_4 into the polynomial gives
which is just a quadratic equation with the two solutions:
The solutions in terms of the original variable are obtained by substituting "x"3 back in for "u", which gives
Then, assuming that one is interested only in real solutions, the solutions of the original equation are
Simple example.
Consider the system of equations
where formula_11 and formula_12 are positive integers with formula_13. (Source: 1991 AIME)
Solving this normally is not very difficult, but it may get a little tedious. However, we can rewrite the second equation as formula_14. Making the substitutions formula_15 and formula_16 reduces the system to formula_17. Solving this gives formula_18 and formula_19. Back-substituting the first ordered pair gives us formula_20, which gives the solution formula_21 Back-substituting the second ordered pair gives us formula_22, which gives no solutions. Hence the solution that solves the system is formula_23.
Formal introduction.
Let formula_24, formula_25 be smooth manifolds and let formula_26 be a formula_27-diffeomorphism between them, that is: formula_28 is a formula_29 times continuously differentiable, bijective map from formula_24 to formula_25 with formula_29 times continuously differentiable inverse from formula_25 to formula_24. Here formula_29 may be any natural number (or zero), formula_36 (smooth) or formula_37 (analytic).
The map formula_28 is called a "regular coordinate transformation" or "regular variable substitution", where "regular" refers to the formula_27-ness of formula_28. Usually one will write formula_41 to indicate the replacement of the variable formula_11 by the variable formula_12 by substituting the value of formula_28 in formula_12 for every occurrence of formula_11.
Other examples.
Coordinate transformation.
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 formula_50 runs outside a formula_51-length interval, for example, formula_52, the map formula_28 is no longer bijective. Therefore, formula_28 should be limited to, for example formula_55. Notice how formula_56 is excluded, for formula_28 is not bijective in the origin (formula_50 can take any value, the point will be mapped to (0, 0)). Then, replacing all occurrences of the original variables by the new expressions prescribed by formula_28 and using the identity formula_60, we get
Now the solutions can be readily found: formula_62, so formula_63 or formula_64. Applying the inverse of formula_28 shows that this is equivalent to formula_66 while formula_67. Indeed, we see that for formula_66 the function vanishes, except for the origin.
Note that, had we allowed formula_56, the origin would also have been a solution, though it is not a solution to the original problem. Here the bijectivity of formula_28 is crucial. The function is always positive (for formula_71), hence the absolute values.
Differentiation.
The chain rule is used to simplify complicated differentiation. For example, consider the problem of calculating the derivative
Writing
we get
Integration.
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. 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.
Differential equations.
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.
Very often, a general form for a change is substituted into a problem and parameters picked along the way to best simplify the problem.
Scaling and shifting.
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 "n"th 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 formula_79 the pressure gradient, both constants. By scaling the variables the problem becomes
where
Scaling is useful for many reasons. 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.
Momentum vs. velocity.
Consider a system of equations
for a given function formula_83.
The mass can be eliminated by the (trivial) substitution formula_84.
Clearly this is a bijective map from formula_85 to formula_85. Under the substitution formula_87 the system becomes
Lagrangian mechanics.
Given a force field formula_89, Newton's equations of motion are
Lagrange examined how these equations of motion change under an arbitrary substitution of variables formula_91, formula_92
He found that the equations
are equivalent to Newton's equations for the function formula_94,
where "T" is the kinetic, and "V" the potential energy.
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.