Calculus: Analytic Geometry and Calculus, with Vectors

(^586) Partial derivatives
and hencef(r + hu) = f(x + hul, y + hue, z + hu3) when u = uli + u2j + uk.
Then, assuming that f has continuous partial derivatives, prove that
(3) D(f, r, u) =
Observe that the scalar components of the vectors r, u, and Of depend upon the
coordinating system which was chosen, but that r, u, D(f, r, u) and Vf do not.
Remark: The intrinsic definition (1) is particularly convenient when "general"
or "abstract" theories are being developed. In fact some parts of modern
mathematics involve "vectors" that are "abstract elements" of "abstract spaces"
for which appropriate axioms are valid. Some elegant theories are developed
without use of coordinate systems. At the other extreme, some elementary
developments of vectors in E3 are tied so rigidly to a single sublime coordinate
system that vectors are identified with ordered sets of numbers (the scalar com-
ponents of the vectors). One virtue of (1) lies in the fact that it can be used
when f is a vector-to-vector function of which the domain and range are both sets
of vectors. Nobody ever learns all about all of these things on a windy Wednes-
day, but people who keep studying mathematics do keep picking up ideas.