Change of variables on arvau

Wikipedia / Change of variables

In mathematics, a change of variables is a basic technique used to simplify problems in which the original variabless 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. Substituting x 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 x3 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

. (Source: 1991 AIME)

Formal introduction

be smooth manifolds-diffeomorphism times continuously differentiable, bijective (smooth (analytic).

is called a regular coordinate transformation or regular variable substitution, where regular.

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

can take any value, the point will be mapped to (0, 0)). Then, replacing all occurrences of the original variables by the new expressions, we get

the function vanishes, except for the origin.

), hence the absolute values.

Differentiation

The chain rule is used to simplify complicated differentiation. For example, consider the problem of calculating the derivative

Then:

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.

Change of variables formula in terms of Lebesgue measure

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. The proof is due to approximations of the Jordan content. diffeomorphism.

Pullback measure and transformation formula

. The change of variables formula for pullback measures is

Pushforward measure and transformation formula

. 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

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

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 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 smaller the number of computations.

Momentum vs. velocity

Consider a system of equations

the system becomes

Lagrangian mechanics

, Newton's equations of motion are

He found that the equations

, 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.