I'm looking for a book with a more practical approach to the content of vector analysis (e.g. smooth manifolds) as calculus is for real analysis.
I give you an example of the a kind of content I'm talking about:
Let $\varphi:V_0\to V$ and $\psi:W_0\to W$ be two parametrizations where $V_0$ and $W_0$ are open sets in $\mathbb R^m$ and $V$ and $W$ are two open sets in $\mathbb R^m$. If $P=\varphi (a)=\psi(b)\in V\cap W$, we have the bases of the tangent vector space $T_pM$ of the manifold $M$ at the point $p:$
$$\text{$\bigg\{\frac{\partial \varphi}{\partial x_1}(a),\ldots,\frac{\partial \varphi}{\partial x_m}(a)\bigg\}$ and $\bigg\{\frac{\partial \psi}{\partial y_1}(b),\ldots,\frac{\partial \psi}{\partial y_m}(b)\bigg\}$}$$
The changing basis matrix is determined by the elements $\alpha_{ij}=\frac{\partial\xi_i}{\partial x_j}(a)$.
I would like to know if there is a book with practical examples and exercises to help me to fix this kind of content.
I'm not sure if it has exactly the kind of examples you're looking for, but the two books that I've seen that have a reasonable number are Munkres 'Analysis on Manifolds' and Hubbard's 'Vector Calculus, Linear Algebra and Differential Forms.' You can probably find both online.