/sci/ - Science & Math

Previously >>12147606

>>12139089
Neither. [math]F(x)[/math] isn't actually a function.
The differential is a map [math]D : C^1 (\Omega) \rightarrow C^0 (\Omega)[/math]. [math]Ker ~ D[/math] is the subspace [math]K ( \Omega) \subset C^{1} (\Omega)[/math] of constant functions. The homomorphism theorem then guarantees that the induced map [math]D : C^{1} (\Omega) / K ( \Omega) \rightarrow C^{0} (\Omega)[/math] is an isomorphism, and has an inverse map [math]\int ~ dx[/math]. Thus, [math]\int f(x) ~ dx = F(x) + C \in C^{1} ( \Omega) / K(\Omega)[/math] is written according to standard quotient notation.