Antidifferentiability (Real Parametric Curves)#

Definition: Antidifferentiability (Real Parametric Curves)

A real parametric curve \(f: \mathcal{D}_{f} \subseteq \mathbb{R} \to \mathbb{R}^n\) is antidifferentiable on a subset \(S \subseteq \mathcal{D}_{f}\) if there exists a real parametric curve \(F: \mathcal{D}_{F} \subseteq \mathbb{R} \to \mathbb{R}^n\) whose derivative on \(S\) is \(f\):

\[F'(t) = f(t) \qquad \forall t \in S\]

Any such \(F\) is known as an antiderivative of \(f\) on \(S\).

Theorem: Component-Wise Antidifferentiability

Let \(f: \mathcal{D}_{f} \subseteq \mathbb{R} \to \mathbb{R}^n\) and \(F: \mathcal{D}_{F} \subseteq \mathbb{R} \to \mathbb{R}^n\) be real parametric curves with component functions \(f_1, \dotsc, f_n\) and \(F_1, \dotsc, F_n\), respectively, and let \(S \subseteq \mathcal{D}_f\).

Then \(F\) is an antiderivative of \(f\) on \(S\) if and only if \(F_1, \dotsc, F_n\) are antiderivatives of \(f_1, \dotsc, f_n\) on \(S\), respectively:

\[F'(t) = f(t) \qquad \forall t \in S \iff F_i'(t) = f_i(t) \qquad \forall t \in S, \forall i \in \{1, 2, \dotsc, n\}\]

Proof

TODO

Theorem: Antiderivatives on Intervals

Let \(f: \mathcal{D}_f \subseteq \mathbb{R} \to \mathbb{R}^n\) be a real parametric curve, let \(I \subseteq \mathcal{D}_f\) be an interval and let \(F: \mathcal{D}_F \subseteq \mathbb{R} \to \mathbb{R}^n\) be an antiderivative of \(f\) on \(I\).

A real parametric curve \(G: \mathcal{D}_G \subseteq \mathbb{R} \to \mathbb{R}^n\) is also an antiderivative of \(f\) on \(I\) if and only if there exists some vector \(\boldsymbol{C} \in \mathbb{R}^n\) such that

\[F(t) = G(t) + \boldsymbol{C}\]

for all \(t \in I\).

Proof

TODO