Partial Antidifferentiability (Real Scalar Fields)#
Definition: Partial Antidifferentiability
Let \(f: \mathcal{D} \subseteq \mathbb{R}^n \to \mathbb{R}\) be a real scalar field.
We say that \(f\) is partially antidifferentiable on an open set \(U \subseteq \mathcal{D}\) with respect to the \(k\)-th variable if there exists a real scalar field \(F: U \subseteq \mathbb{R}^n \to \mathbb{R}\) whose partial derivative w.r.t. the \(k\)-variable is \(f\):
\[\partial_k F = f\]
We call \(F\) a partial antiderivative of \(f\) with respect to the \(k\)-th variable.
Notation
If the components of the input are labelled \(x_1, \dotsc, x_n\), i.e. \(f(x_1, \dotsc, x_n)\), then the set of all partial antiderivatives of \(f\) on \(U\) w.r.t. the \(k\)-th variable is denoted as follows:
\[\int f(x_1, \dotsc, x_n) \,\mathrm{d}x_k\]
Theorem: Antiderivatives Structure
TODO
Proof
TODO