Definition:Derivative/Vector-Valued Function/Open Set
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Definition
Let $\mathbf r: t \mapsto \map {\mathbf r} t$ be a vector-valued function defined for all $t$ on some real interval $\mathbb I$.
The derivative of $\mathbf r$ with respect to $t$ is defined as the limit:
\(\ds \map {\mathbf r'} t\) | \(:=\) | \(\ds \lim_{\Delta t \mathop \to 0} \frac {\map {\mathbf r} {t + \Delta t} - \map {\mathbf r} t} {\Delta t}\) |
for all $t$ for which the limit exists.
Sources
- 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text {III}$: The Differentiation of Vectors: $1$. Scalar Differentiation
- 2005: Roland E. Larson, Robert P. Hostetler and Bruce H. Edwards: Calculus (8th ed.): $\S 12.2$
- For a video presentation of the contents of this page, visit the Khan Academy.