Maths - Partial Differentiation

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

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Maths - Partial Differentiation

Small Increments:

Rates of Change:

Implicit functions:

Change of variables

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