# 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 `x=y+1`, the expression `z=5(y+1)` could be rewritten `z=5x`. 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 `u=x ^{2}` and

`v=1/x`, 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 for g'(x).

### Multiple integrals

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.