Differentiation of Vector-Valued Function Componentwise

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Theorem

Let:

$\mathbf r: t \mapsto \tuple {\map {r_1} t, \map {r_2} t, \ldots, \map {r_n} t}$

be a differentiable vector-valued function.


The derivative of a vector-valued function can be calculated by differentiating each of its component functions:

$\dfrac {\d \map {\mathbf r} t} {\d t} = \tuple {\dfrac \d {\d t} \map {r_1} t, \dfrac \d {\d t} \map {r_2} t, \ldots, \dfrac \d {\d t} \map {r_n} t}$


Proof

\(\ds \frac {\d \map {\mathbf r} t} {\d t}\) \(=\) \(\ds \lim_{\Delta t \mathop \to 0} \frac {\map {\mathbf r} {t + \Delta t} - \map {\mathbf r} t} {\Delta t}\) Definition of Derivative of Vector-Valued Function
\(\ds \) \(=\) \(\ds \begin {bmatrix} \ds \lim_{\Delta t \mathop \to 0} \dfrac {\map {r_1} {t + \Delta t} - \map {r_1} t} {\Delta t} \\ \ds \lim_{\Delta t \mathop \to 0} \dfrac {\map {r_2} {t + \Delta t} - \map {r_2} t} {\Delta t} \\ \vdots \\ \ds \lim_{\Delta t \mathop \to 0} \dfrac {\map {r_n} {t + \Delta t} - \map {r_n} t} {\Delta t} \end {bmatrix}\) Definition of Limit of Vector-Valued Function
\(\ds \) \(=\) \(\ds \begin {bmatrix} \dfrac \d {\d t} \map {r_1} t \\ \dfrac \d {\d t} \map {r_2} t \\ \vdots \\ \dfrac \d {\d t} \map {r_n} t \end {bmatrix}\) Definition of Derivative of Real Function

$\blacksquare$


Sources

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