Definition:Definite Integral of Vector-Valued Function
Jump to navigation
Jump to search
Definition
Let $I = \closedint a b \subset \R$ be a closed real interval.
Let $\mathbf f: I \to \R^n$ be a vector-valued function on $I$:
- $\forall x \in I: \map {\mathbf f} x = \ds \sum_{k \mathop = 1}^n \map {f_k} x \mathbf e_k$
where:
- $f_1, f_2, \ldots, f_n$ are real functions from $U$ to $\R$
- $\tuple {e_1, e_2, \ldots, e_k}$ denotes the standard ordered basis on $\R^n$.
Let $\mathbf f$ be differentiable on $I$.
Let $\map {\mathbf g} x := \dfrac \d {\d x} \map {\mathbf f} x$ be the derivative of $\mathbf f$ with respect to $x$.
The definite integral of $\mathbf g$ with respect to $x$ from $a$ to $b$ is defined as:
- $\ds \int_a^b \map {\mathbf g} x \rd x := \map {\mathbf f} b - \map {\mathbf f} a$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 22$: Integrals involving Vectors: $22.47$