 
Summary: The differential.
Let n be a positive integer.
Suppose f is functon whose domain is a subset of Rn
and which has values in Rm
for some positive
integer m. For each j = 1, . . ., n the partial derivative
jf
is, by definition, the set of ordered pairs (a, v) such that a is an interior point of the domain of f and
v = lim
h0
1
h
f(a + hej)  f(a) .
Note that for each j = 1, . . ., n the partial derivative jf is a function, possibly empty, with values in Rm
whose domain is a subset of the domain of f.
In case n = 1 we set
f (a) = 1f(a).
We say f is differentiable at a if a is in the domain of each of the partial derivatives jf, j = 1, . . ., n
and if
lim
