Explanation of theorem on differential forms

Question

This is text from Do Carmo's Differential forms and applications Page-10. Could anyone explain the highlighted step?If f* is applied to each of the term then how do we get the RHS of the highlighted equation as shown in the figure. Thanks.

Answer 1

In general for $h(y) dy^i$, we have

$$f^*(h(y) dy^i) = (f^* h)(x) df^i = \sum_j h(f(x)) \frac{\partial f^i}{\partial x^j} dx^j = \sum_j h \frac{\partial f^i}{\partial x^j} dx^j.$$

The last step is nothing but suppressing that $f(x)$ from $h(f(x))$ (just notational convenience). Thus

$$\begin{split} f^*\left( \sum_i \frac{\partial g}{\partial y^i} dy^i \right) &= \sum_i f^*\left( \frac{\partial g}{\partial y^i} dy^i \right)\\ &= \sum_i \left( \frac{\partial g}{\partial y^i} \sum_j \frac{\partial f^i}{\partial x^j} dx^j \right) \\ &=\sum_{i,j} \left( \frac{\partial g}{\partial y^i} \frac{\partial f^i}{\partial x^j} dx^j \right) \end{split}$$