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The differential. Let n be a positive integer.
 

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

  

Source: Allard, William K. - Department of Mathematics, Duke University

 

Collections: Mathematics