View source for Change of variables

'''Changing variables''' is a mathematical practice used to put equations into more convenient forms by substituting certain variables for expressions which they are equal to.  For example, if <tt>x=y+1</tt>, the expression <tt>z=5(y+1)</tt> could be rewritten <tt>z=5x</tt>.  Change of variables is never absolutely necessary, but it can greatly simplify problems and help avoid errors.  While most frequently used in [[integration]], it can also be helpful to a lesser degree in [[algebra]].

A change of variables transforms the original plain into another.  Areas and lengths are transformed in a uniform ratio, allowing one to solve problems in the second plain and then transform the areas back to the original plane.  For example, when one defines <tt>u=x<sup>2</sup></tt> and <tt>v=1/x</tt>, the x-y plane is transformed into the u-v plane.

==Uses==
===[[Chain rule]]===
Changing variables can be quite useful in differentiating according to the chain rule.  Since the derivative of f(g(x)) = f'(g(x))*g'(x), it might be helpful to substitute a single variable for g(x) before differentiating.

===U substitution===
When integrating a derivative formed according to the chain rule, one first needs to identify g'(x) and separate it from f'(g(x)).  To avoid confusion, it can often be helpful here to substitute a single variable (traditionally "u" or "&mu;") for g(x) and <math>\frac{du}{dx}</math> for g'(x).

===[[Integral#Multiple integrals|Multiple integrals]]===
[[Image:Parabola.png|right|thumb|pxl=200|Change of variables can help one integrate over irregular regions, such as the areas under and between these parabolas]]

Changing of variables can help one integrate over irregular regions by turning the irregular boundaries into straight lines.  For instance, someone can transform 1/x into the straight line v=1/x.  This can be quite helpful when integrating regions with multiple curved borders.


[[Category:Calculus]]
[[Category:Integration]]