## Small Increments:

If z=f(x,y) then d z = | ∂ z | d x + | ∂ z | d y |

∂ x | ∂ y |

## Rates of Change:

We can express the above expression as a rate of change with respect to some extra dimension 't'.

d z | = | ∂ z | d x | + | ∂ z | d y |

d t | ∂ x | d t | ∂ y | d t |

## Implicit functions:

Or if we use one of the existing dimensions such as 'x':

d z | = | ∂ z | + | ∂ z | d y |

d x | ∂ x | ∂ y | d x |

## Change of variables

Or we can change to a new set of variables (u,v):

∂ z | = | ∂ z | ∂ x | + | ∂ z | ∂ y |

∂ u | ∂ x | ∂ u | ∂ y | ∂ u |

∂ z | = | ∂ z | ∂ x | + | ∂ z | ∂ y |

∂ v | ∂ x | ∂ v | ∂ y | ∂ v |