Constants

What should we add a constant after getting the Antiderivative of first order functions?

Few days ago, I saw this question under @Phenomene Bizarre‘s post 如何正确理解群论中的同态基本定理？. I thought this different approach to explain a simple question is really interesting. So here comes this blog.

Background

First we should clarify the question a little bit. As asked, when we try to find the antiderivative of a first order function $f$ with respect to variable $x$, $$ \int f \ dx = F + c. $$ why should we always add the constant $c$ at the end?

Definitions

Definition. $C^1(\mathbb{R})$ is the set of first order continuously differentiable functions on $\mathbb{R}$.

An Algebraic Approach

To talk about antiderivative, we should first understand the definition of derivative. As defined, derivative of a function in this case is epimorphism mapping from $C^1(\mathbb{R})$ to $C^0(\mathbb{R})$, $$ D := \frac{d}{dx} : C^1(\mathbb{R}) \rightarrow C^0(\mathbb{R}). $$

That is, if a function $f$ nis integrable, then it must be in the image of the above function, $$ f \text{ is integrable} \iff f \in \mathrm{im} D. $$

Since $D$ is an epimorphism, by the First Isomorphism Theorem, there exists an unique isomorphism $$ \phi : C^1(\mathbb{R}) \big/ \ker D \rightarrow \mathrm{im}D $$ where $C^1(\mathbb{R}) \big/ \ker D$ is the quotient space of $C^1(\mathbb{R})$.

Since isomorphism is bijective, there exists an inverse function of $\phi$ $$ \phi^{-1} : \mathrm{im}D \rightarrow C^1(\mathbb{R}) \big/ \ker D $$ that maps form the image of function $D$ to the quotient space, which as we know is the antiderivative function. Then it is clear that the antiderivative of $f$, $F \in C^1(\mathbb{R}) \big/ \ker D$.

Since $\ker D$ is a normal subgroup to $C^1(\mathbb{R})$, by definition of quotient group, $C^1(\mathbb{R}) \big/ \ker D$ is the set of all left cosets of $\ker D$ in $C^1(\mathbb{R})$. That is, $$ C^1(\mathbb{R}) \big/ \ker D = \{ a + \ker D: a \in C^1(\mathbb{R}) \}. $$

Here we say it is $a + N$ because $+$ is the binary operation defined on $C^1(\mathbb{R})$.

Then what is the kernel of function $D$? By definition, the kernel of a morphism is the set of all elements from the domain mapped to the identity element in image. In this case, since the identity of $+$ on $\mathbb{R}$ is $0$, $$ \ker D = \{a : a \in C^1(\mathbb{R}) \land a \mapsto 0 \} \subseteq \mathbb{Z}. $$

Therefore, we should add the constant when finding antiderivative or indefinite integral.