I don't follow Loring Tu's use of the chain rule in a proof to problem 8.9 (Introduction to Manifolds). I'll only focus on the relevant excerpt.
Let $(V, y^1, \ldots, y^n)$ by a chart about $p$ in a manifold of dimension $n$. Define a new coordinate system $x^1, \ldots x^n$ by
$$
y^i=\sum_{j=1}^n a^i_j x^j ~~~~~~i= 1, \ldots n
$$
Then by the chain rule,
$$
\frac{\partial }{\partial x^j} = \sum_i \frac{\partial {y^i}}{\partial x^j} \frac{\partial }{\partial y^i} \tag{1}
$$
I see in this post the author called this use of the chain rule the "basis transformation rule".
Can someone provide an intuitive explanation for why (1) is true?
Best Answer
Every tangent vector $v \in T_x M$ can be written as $$v = a^i \left.\frac{\partial}{\partial y^i} \right\vert_{x} $$
where $a_i = v(y^i)$, since $$v(y^j) = a^i\left.\frac{\partial y^j}{\partial y^i} \right\vert_{x} = a^i \delta_{i}^j = a_j $$ So, taking $v = \frac{\partial}{\partial x^j}$, we see that:
$$\left.\frac{\partial}{\partial x^j} \right\vert_{x} = \left(\left.\frac{\partial }{\partial x^j} \right\vert_{x} \right)(y^i) \left.\frac{\partial}{\partial y^i} \right\vert_{x} = \left.\frac{\partial y^i}{\partial x^j} \right\vert_{x} \left.\frac{\partial}{\partial y^i} \right\vert_{x} $$