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'''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 "μ") 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]]
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 "μ") 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]]