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When differentiating functions of several variables which
are called scalar fields, the gradient of the function, contains
the all the information about the tangent (hyper)-plane (or
linear approximation or differentials) that there is.
When differentiating vector fields, which are vector
valued functions of several variables, there are partial derivatives
of every component in every direction; thus there are nine
of them in three dimensions. However the divergence
and curl are the combinations of these that are particular
importance in applications, and we concentrate our attention
on them. These are written as
with
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