Change of variables

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 new ones; the new and old variables being related in some specified way. The intent is that the problem expressed in new variables may be simpler, or equivalent to a better understood problem.

A very simple example of a useful variable change can be seen in the problem of finding the roots of the sixth order polynomial:

x^6 - 9 x^3 + 8 = 0 \,

Sixth order polynomial equations are generally impossible to solve in terms of elementary functions. This particular equation, however, may be simplified by defining a new variable x3 = u. Substituting this into the polynomial:

u^2 - 9 u + 8 = 0 \,

which is just a quadratic equation with solutions:

u = 1 \quad \mbox{and} \quad u = 8

The solution in terms of the original variable is obtained by replacing the original variable:

x^3 = 1 \quad \mbox{and} \quad x^3 = 8 \quad \Rightarrow \qquad x =  (1)^{1/3} = 1 \quad \mbox{and} \quad x =  (8)^{1/3} = 2\,



The chain rule is used to simplify complicated differentiation. For example, to calculate the derivative

\frac{d}{d x}\left(\sin(x^2)\right)\,

the variable x may be changed by introducing x2 = u. Then, by the chain rule:

\frac{d}{d x} = \frac{d}{d u} \frac{d u}{d x} = \frac{d}{d x}\left(u\right) \frac{d}{d u} = \frac{d}{d x}\left(x^2\right) \frac{d}{d u} = 2 x \frac{d}{d u}\,

so that

\frac{d}{d x}\left(\sin(x^2)\right) = 2 x \frac{d}{d u}\left(\sin(u)\right) = 2 x \cos(x^2)\,

where in the very last step u has been replaced with x2.


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.

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 nth order derivative, the change simply results in

\frac{d^n y}{d x^n} = \frac{y_\text{scale}}{x_\text{scale}^n} \frac{d^n \hat y}{d \hat x^n}


x = \hat x x_\text{scale} + x_\text{shift}
y = \hat y y_\text{scale} + y_\text{shift}

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

\mu \frac{d^2 u}{d y^2} = \frac{d p}{d x} \quad ; \quad u(0) = u(\delta) = 0

describes parallel fluid flow between flat solid walls separated by a distance δ; µ is the viscosity and dp / dx the pressure gradient, both constants. By scaling the variables the problem becomes

\frac{d^2 \hat u}{d \hat y^2} = 1 \quad ; \quad \hat u(0) = \hat u(1) = 0


y = \hat y \delta \qquad \mbox{and} \qquad u = \hat u \frac{L^2}{\mu} \frac{d p}{d x}

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.

See also

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