Definition:Differential of Mapping/Vector-Valued Function/Point
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Definition
Let $U \subset \R^n$ be an open set.
Let $f: U \to \R^m$ be a vector-valued function.
Let $f$ be differentiable at a point $x \in U$.
The differential of $f$ at $x$ is the linear transformation $\d \map f x: \R^n \to \R^m$ defined as:
- $\map {\d \map f x} h = \map {J_f} x \cdot h$
where:
- $\map {J_f} x$ is the Jacobian matrix of $f$ at $x$.