what is a univariate constant?

Question

I've come across a constant so called 'univariate constant'. The one obtained when integrating a partial derivative such as C(y). Is is like a function?

Answer 1

Assume you have a function of one variable, $f(x)$, and you want to determine a function $F(x)$ such that $F(x)'=f(x)$. You will notice that if $F_0(x)$ is a solution, then $F_0(x)+C$ is one as well, for any value of $C$. Thus one says that the antiderivative of $f(x)$ is $F(x)+C$.

In higher dimensions, if you are given $f(x,y)$ and you want to determine a function $F(x,y)$ such that $\frac{d}{dx}F(x,y)=f(x,y)$ for all $x$ and $y$, you will similarly see that if $F_0(x,y)$ is a solution, then $F_0(x,y)+C(y)$ is one as well, no matter how you choose the function $C(y)$. Thus, the word constant that your professor uses is based on the equivalent analysis in one dimensions, but you are right that one should instead better speak of a function that is independent of $x$, or a function that is constant in $x$.