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Diﬀerential Calculus for Real-Valued
Functions of Several Variables
In this chapter and in Chapter 5, we consider functions whose input
involves several variables—or equivalently, whose input is a vector—and
whose output is a real number.
We shall restrict ourselves to functions of two or three variables, where the
vector point of view can be interpreted geometrically.
A function of two (resp. three) variables can be viewed in two slightly
diﬀerent ways, reﬂected in two diﬀerent notations.
We can think of the input as three separate variables; often it will be
convenient to use subscript notation xi (instead of x, y, z ) for these
variables, so we can write
f (x, y ) = f (x1 , x2 )
in the case of two variables and
f (x, y, z ) = f (x1 , x2 , x3 )
in the case of three variables.
Alternatively, we can think of the input as a single vector − formed from
listing the variables in order:
− = (x, y ) = (x , x )
1 215 2 ...
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