Definition:Differential of Mapping/Real Function/Open Set

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Definition

Let $U \subset \R$ be an open set.

Let $f : U \to \R$ be a real function.

Let $f$ be differentiable in $U$.

The differential $\d f$ is the mapping $\d f : U \to \map {\operatorname {Hom} } {\R, \R}$ defined as:

$\map {\paren {\d f} } x = \map {\d f} x$

where:

$\map {\d f} x$ is the differential of $f$ at $x$

$\map {\operatorname {Hom} } {\R, \R}$ is the set of all linear transformations from $\R$ to $\R$.

The differential $\d f$ can be regarded as a (real) function of two variables, defined as:

$\map {\d f} {x; h} = \map {f'} x h$

where $\map {f'} x$ is the derivative of $f$ at $x$.