Submersion or not?

Question

How can I prove that $$ f : \mathbb R \rightarrow \mathbb R, f(x) = e^x$$ is a submersion? I know that I must prove that $$f’= e^x$$ is everywhere surjective. But how can I write it? Thank you in advance!

Answer 1

Following the definition, we have to show that for all $x\in \mathbb R$ the differential: \begin{equation} df_x:T_x\mathbb R\rightarrow T_{f(x)}\mathbb R \end{equation} Since $T_x\mathbb R = \mathbb R$ we have only have to show that, fixed $x\in \mathbb R$, the linear application \begin{equation} df_x:\mathbb R\longrightarrow \mathbb R, \quad v\longmapsto df_x(v) = f'(x)\cdot v = e^x\cdot v \end{equation}

So the only thing you have to check is that for all $x\in \mathbb R$, $df_x$ is not the zero map, and this is true because $e^x>0$. Then $f$ is a submersion.