Definition:Derivative/Vector-Valued Function/Point
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Definition
Let $U \subset \R$ be an open set.
Let $\map {\mathbf f} x = \ds \sum_{k \mathop = 1}^n \map {f_k} x \mathbf e_k: U \to \R^n$ be a vector-valued function.
Let $\mathbf f$ be differentiable at $u \in U$.
That is, let each $f_j$ be differentiable at $u \in U$.
The derivative of $\mathbf f$ with respect to $x$ at $u$ is defined as
- $\map {\dfrac {\d \mathbf f} {\d x} } u = \ds \sum_{k \mathop = 1}^n \map {\dfrac {\d f_k} {\d x} } u \mathbf e_k$
where $\map {\dfrac {\d f_k} {\d x} } u$ is the derivative of $f_k$ with respect to $x$ at $u$.
Also see
- Differentiation of Vector-Valued Function Componentwise: justification for this definition
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 22$: Derivatives of Vectors: $22.22$