Show $f^*dx_i = \sum_{j=1}^l \frac{\partial f_i}{\partial y_j} dy_j = df_i$

I'd like to point out, before trying to answer this question, that your definitions confuse me a little. For instance, what is $I(y)$? And in your expression for $\omega$,

$\omega = \sum_{1 \le i_1 < . . . i_k \le n}I(y)dx_i$,

why is there apparently a multi-index of some sort on the $\Sigma$ symbol which doesn't seem (to me at least) to occur in the summand $I(y)dx_i$? Well, perhaps this stuff is explained in Guillemin and Pollack, which I haven't looked at in quite awhile, fine book though it be. Not trying to be critical here, merely seeking clarification.

Having said these things, let's try to prove that

$f^*dx_i = \sum_{j = 1}^l\frac{\partial f_i}{\partial y_j}dy_j = df_i$.

I'm going to try to do this the way I learned it, mostly from general principles; as I indicated above, I don't have a copy of Guillemin and Pollack in front of me, so if I seem like I'm winging it, bear with me . . .

The first thing you need to know is that $f^*$ is the adjoint of the map $f_*$, in the sense usually used in linear algebra: if $T:V \to W$ is a linear map between vector spaces $V$ and $W$, then for any $\sigma \in W^*$ we define the linear functional $T^*\sigma \in V^*$ by the formula $T^*\sigma(v) = \sigma(Tv)$ for vectors $v \in V$. Thus it is easily seen that $T^*:W^* \to V^*$ is also a linear map. This idea of course applies pointwise with respect to $U$ and $V$, i.e. fiberwise with respect to the tangent and cotangent spaces of $U$ and $V$. Now let's look at $f^*dx_i$. For any tangent vector $Y \in T_yU$, we have

$(f^*dx_i)(Y) = dx_i(f_*(Y))$,

and with

$Y = \sum_{j = 1}^l Y^j\frac{\partial}{\partial y^j}$

we obtain

$(f^*dx_i)(Y) = (f^*dx_i)(\sum_{j = 1}^l Y^j\frac{\partial}{\partial y^j})$,

and by linearity of everything this yields

$(f^*dx_i)(Y) = \sum_{j = 1}^lY^j (f^*dx_i)(\frac{\partial}{\partial y^j}) = \sum_{j = 1}^lY^jdx_i(f_*(\frac{\partial}{\partial y^j}))$.

We scrutinize $f_*(\frac{\partial}{\partial y^j})$. With $f = (f_1, f_2, . . . f_k)$, with each $f_p$, $1 \le p \le k$ being a function of the $y_q$, $1 \le q \le l$, and $g:U \to R$ and sufficiently differentiable, the vector field $f_*(\frac{\partial}{\partial y^j})$ on $U$ may be applied to $g$:

$f_*(\frac{\partial}{\partial y^j})[g(x_1, x_2, . . . x_k)] = \frac{\partial}{\partial y^j}(g(f_1(y_1, . . . y_l), f_2(y_1, . . . y_l), . . . f_k(y_1, y_2, . . . , y_l)))$

$= \sum_{n = 1}^k \frac{\partial g}{\partial x_n} \frac{\partial f_n}{\partial y_j}$,

this last equality following from the fact that $x_p = f_p(y_1, y_2, . . . , y_l)$ and the chain rule. Thus we see that the vector field $f_*(\frac{\partial}{\partial y^j})$ satisfies

$f_*(\frac{\partial}{\partial y^j}) = \sum_{n = 1}^k \frac{\partial f_n}{\partial y_j}\frac{\partial}{\partial x_n} $,

and if this is inserted into our previous expression for $(f^*dx_i)(Y)$,

$(f^*dx_i)(Y) = \sum_{j = 1}^lY^jdx_i(f_*(\frac{\partial}{\partial y^j}))$,

it follows that

$(f^*dx_i)(Y) = \sum_{j = 1}^l \sum_{n = 1}^k Y^j dx_i(\frac{\partial f_n}{\partial y_j}\frac{\partial}{\partial x_n})$,

and using $dx_i(\frac{\partial}{\partial x_n}) = \delta_{in}$,

$(f^*dx_i)(Y) = \sum_{j = 1}^l Y^j \frac{\partial f_i}{\partial y_j} = \sum_{j = 1}^l \frac{\partial f_i}{\partial y_j}dy_j(Y) = df_i(Y)$,

since we have $Y = \sum_{j = 1}^l Y^l \frac{\partial}{\partial y_l}$ and $dy_j(Y) = Y^j$. Since this holds for any $Y \in T_yV$, we have shown that

$f^*dx_i = \sum_{j = 1}^l \frac{\partial f_i}{\partial y_j}dy_j = df_i$,

as per request. QED.

Whew! Too many indices and subscripts to keep track of!

Now, as I recall, establishing this formula on $1-$forms allows it to be extended in the usual manner to all of $\Lambda(T*U)$, i.e., all form (fields) by careful use of the definitions of the $\wedge$ product and a lot of maneuvering of matrices and indices.

Gotta run, my night job beckons.

We know $f^*dx_i$ is tensor field and compute it at $p\in V$ which is arbitary,$(f^*{dx_i})_p$=${dx_i}_{f(p)}{f_*}_p$(from definition of pullback),also we have ${f_*}_p=df_p=\Big[\frac{\partial f_t}{\partial y_j}(p)\Big]_{k\times l},$ and ${dx_i}_{f(p)}={{x_i}_*}_{f(p)}=[0,...,0,1,0,...,0]_{1\times k}$(with 1 in ith place),the multiplication of two matrices is $\Big[\frac{\partial f_i}{\partial y_j}(p)\Big]_{1\times l}$ which is equal to ${df_i}_p$,and ${df_i}_p=\sum_{j=1}^l \frac{\partial f_i}{\partial y_j}(p){ dy_j}_p$.

First, I think both answers offered thus far are excellent in their own way. I will merely attempt to say the same with a slightly different notation.

First, $f=(f^1, \dots, f^k)$ is a function of $y = (y^1, \dots , y^l)$. Therefore, we differentiate $f$ with respect to $y$ at the point $y=p$ we have: $$ df_p(h) = \sum_{i=1}^k\sum_{j=1}^l[f'(p)]_{ij}h_j\frac{\partial}{\partial x^i}\bigg|_{f(p)} \qquad \text{where} \ [f'(p)]_{ij} = \frac{\partial f^i}{\partial y^j} $$ where $f'(p) \in \mathbb{R}^{k \times l}$ and $[h_j] \in \mathbb{R}^l$ thus $f'(p)[h] \in \mathbb{R}^k$. I suppose, to be explicit, $h = \sum_{j=1}^l h_j\frac{\partial}{\partial y^j}\bigg|_p$. Here I, as is my custom, assume $\mathbb{R}^n = \mathbb{R}^{1 \times n}$, that is, euclidean space is made of column vectors. I'll place $dx^j$ at $f(p)$ but I'll forego adorning $dx^j$ with that point-dependence for brevity in what follows. The pull-back of $dx^j$ from the range of $f$ will form (pun-intended) a one-form $\alpha$ in the domain at $p$. We can calculate such a one-form by calculating its components in the $\{dy^1, \dots , dy^l \}$ basis for $T^*V_p$ by evaluation at the coordinate vector fields to which these are dual. Consider, $$ \bigl((f^*)_{p}(dx^m)\bigr)(\frac{\partial}{\partial y^n}\bigg|_p) = dx^m ( df_p(\frac{\partial}{\partial y^n}\bigg|_p)) \qquad \star$$ Now, $h_j = \delta_{jn}$ for the coordinate vector field ($h = \sum_{j=1}^{l}\delta_{jn}\frac{\partial}{\partial y^n}\bigg|_p)$ hence $$ df_p(\frac{\partial}{\partial y^n}\bigg|_p)) = \sum_{i=1}^k\sum_{j=1}^l[f'(p)]_{ij}\delta_{jn}\frac{\partial}{\partial x^i}\bigg|_{f(p)} = \sum_{i=1}^k[f'(p)]_{in}\frac{\partial}{\partial x^i}\bigg|_{f(p)} $$ Finally, feed this into the $\star$ equation, \begin{align} \bigl((f^*)_{p}(dx^m)\bigr)(\frac{\partial}{\partial y^n}\bigg|_p) &= dx^m (\sum_{i=1}^k[f'(p)]_{in}\frac{\partial}{\partial x^i}\bigg|_{f(p)}) \\ &= \sum_{i=1}^k[f'(p)]_{in} dx^m (\frac{\partial}{\partial x^i}\bigg|_{f(p)}) \\ &= \sum_{i=1}^k[f'(p)]_{in}\delta_{im} \\ &= [f'(p)]_{mn} \end{align} Therefore, $$ (f^*)_{p}(dx^m) = \sum_{n=1}^l [f'(p)]_{mn} dy^n = \sum_{n=1}^l \frac{\partial f^m}{\partial y^n}dy^n = df^m$$