Antiderivative of a Vector-Valued Function

Topics: Vector-Valued Function - Derivative of a Vector-Valued Function

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(definition)

Let $r:{\mathcal{D}}_1\subseteq \mathbb{R}\to V_n$ be a vector-valued function and $r^{\prime }:{\mathcal{D}}_2\subseteq \mathbb{R}\to V_n$ its derivative.

We will define the following function as the antiderivative (or primitive function) of $r^{\prime }$:

$$\int r^{\prime }dt:=r+\vec{c}$$

Note that $\vec{c}\in V_n$ (it’s a vector).

(theorem)

Given $r(t)=⟨r_1(t),\dots ,r_n(t)⟩$, then:

$$\int rdt=⟨\int r_{1}dt,\dots ,\int r_{n}dt⟩$$