Well, first of all what is the differential of a function
the differential is a local concept; you have to fix a point $x$ in the domain and the differential is a linear map that send the tangent space $T_x R$ in $T_{f(x)}R$
you mean "for all point $x$ the differential at point $x$ is a surjective map"
so you fave to fix $x$ and the differential become a linear map from $T_x R$ to $T_f(x)R$
in our example take $x=1$
you have to show that the differential at point $x_1$ that goes from $T_1 R$ to $T_e R$ is surjective
and following the definition you discover that the differential at point $x=1$ is just a linear map that send an element $v\in T_1 R = R$ to the element $e^1 \cdot v$ that lives in $T_f(1) R = R$
Ok, My question is here, if I choose 0 in $T-f(x)$ then I can say that f is not a surjective map because there is not any “x” in domain that “e^x is not equal to 0”. What is the difference between these? The one you told me for “df “ and the one that Isaid now?
So as a conclusion,If I”m not mistaken in my question The appearance in “f(x)” and “df(x) is the same but it is an submersion which is not surjective, it is a general statement,am I right?
I mean, When you ask a question in Stack and people wrote answers, you can accept one of them as the "full answer" to your question. You can do that in your question "submersion or not" page.
For the chat no problem, we can leave and there is no problem
other people will use this chat for speaking about differetial geometry
I told you that because i see in your profile that you forget to accept answers
We know that for any distribution W in TM, one can associate the Lagrangian subbundle. In this case how can I prove this statement?"This subbundle is dirac if and only if W is involutive."
