Vector analysis book with more practical examples and exercises

Question

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.

Answer 1

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.