So I recently started studying smooth manifold theory. I understood the definition of a manifold. It is basically union of sets homeomorphic to open sets in $\mathbb{R}^n$. Now I learnt about cotangent spaces, tangent spaces, induced map between tangent spaces of two smooth manifolds $T_x M $ and $T_x N$ by a smooth map $f:M \to N$. Then I got to know about immersion and submersions.
But I am unable to get the real intution about them. I mean, What would these terms be, when we consider the euclidean spaces? Can anyone help me with that? Is there any references from where I can make sure that the basic is clear? I would request someone to explain me the above terms in the euclidean spaces. It would be helpful if someone can suggest a basic book for understanding purpose. Right now I am following "Differentiial Topology by Morris W. Hirsch" and "Foundations of Differential Manifolds and Lie Groups by Frank Warner"
In our class, the cotangent space have been defined as the quotient $\frac{\operatorname{Germ}_x}{m_x \cap S_x}$ and the tangent space is the dual of the above vector space. Here $m_x$ denote the ideal of $\operatorname{Germ}_x$ such that the function vanishes at $x$. The $S_x$ is the stationary germs, i.e the first partial deriviatives vanish.