Differential Form Pullback Definition I'm having some difficulty following how Spivak (Calculus on Manifolds) has induced the pullback on page 89. 
From reading elsewhere online it seems convention is to define the induced map of the pushforward of a differentiable function $f: \mathbb R^n \to \mathbb R^m$ with the corresponding linear transformation $Df(p): \mathbb R^n \to \mathbb R^m$ as $$f_*: \mathbb R^n_p \to\mathbb R^m_{f(p)}$$$$f_*(v_p) = (Df(p)(v))_{f(p)}$$ and to then use this definition for the pullback, defined as $$f^*:\Lambda(\mathbb R^m_{f(p)})\to \Lambda(\mathbb R^n_p)$$$$f^*\omega(p)(v_1, .., v_k) = \omega(f(p))(f_*(v_1),..., f_*(v_k)),$$where $\omega$ is a k-form on $\mathbb R^m.$
However Spivak has offered the induced definition for the pullback as $$(f^*\omega)(p) = f^*(\omega(f(p))).$$ which then leads to the above definition. 
I'd be grateful for any help explaining the intuition behind this.
 A: After further study of Spivak and other texts, I take back my original comment that there is a typo.  This is an example where there is technically no error, but poor notation and Spivak's famous terseness makes the book extremely confusing for students.  
On page 77, Spivak defines the pullback of a tensor $T\in\mathcal{T}^k(W)$ by a linear transformation $f:V\to W$ as $f^*T(v_1,\ldots,v_k)=T(f(v_1),\ldots, f(v_k))$.
On page 89, he defines $f_*:\mathbb{R}^n_p\to \mathbb{R}^m_{f(p)}$ to be the differential of smooth map $f:\mathbb{R}^n\to\mathbb{R}^m$ as $f_*(v_p)=(Df(p)(v))_{f(p)}$.  Given this definition, we can pull back the $\it{value}$ of a differential form $\omega$ at $f(p)$, $\omega(f(p))\in\mathcal{A}^k(\mathbb{R}^m_{f(p)})$ (which is an alternating tensor on $\mathbb{R}^m_{f(p)}$), by $f_*$, by defining $(f_*)^*(\omega(f(p)))$ for $v_{1p},\ldots, v_{kp}
\in \mathbb{R}^n_p$ as 
$$[(f_*)^* (\omega(f(p)))](v_{1p},\ldots,v_{kp})=
[\omega(f(p))](f_*(v_{1p}),\ldots,f_*(v_{kp})),$$
as an application of the definition on p. 77.
Note that $(f_*)^* (\omega(f(p)))\in\mathcal{A}^k(\mathbb{R}^n_p)$.  Thus, $(f_*)^*$ is a map $\mathcal{A}^k(\mathbb{R}^m_{f(p)})\to \mathcal{A}^k(\mathbb{R}^n_p)$.    This is the induced linear transformation $f^*$ that Spivak talks about.
What Spivak fails to mention is that $f^*\equiv (f_*)^*$ is the conventional simplified notation of this map $(f_*)^*$.  The confusion comes from the fact that the reader was introduced to the related but different definition of $f^*$ on p. 77.  Baffling to the novice though accurate, Spivak introduces $f^*$ as the map induced by $f_*$ without mentioning its relation to the p. 77 definition or this notational convention.
If this is still confusing, the preceding discussion can be ignored: the definition of $f^*$ useful in practice is the one that he gives at the top of p. 90:
$$f^*\omega(p)(v_{1p},\ldots, v_{kp})=\omega(f(p))(f_*(v_{1p}),\ldots,
f_*(v_{kp})).$$
Finally, what Spivak means by $(f^*\omega)(p)=f^*(\omega(f(p)))$ is that he is defining a new differential form, $f^*\omega$ by $\it{pointwise}$ pullback of each tensor $\omega(f(p))$.  Just to be clear, for each point $p$, $(f^*\omega)(p)\in\mathcal{A}^k(\mathbb{R}^n_p)$, so that $f^*\omega$ is a $k$-form on $\mathbb{R}^n$, while $\omega$ is a $k$-form on $\mathbb{R}^m$.
For detailed discussions of definitions, see: Volume 1 of A Comprehensive Introduction to Differential Geometry by Spivak (where all the confusing things in Calculus on Manifolds are actually explained by the author himself) or An Introduction to Manifolds by Loring Tu, which is probably the clearest introduction to smooth manifold theory for first-year graduate students and advanced undergrads.
PREVIOUS REPLY:
There's a typo on p. 89: the definition of the pullback of a differential form $\omega$ should be $(f^*\omega)(p)=(f_*)^* (\omega(f(p)))$.  The rhs of this definition is based on the definition of the pullback of a tensor, given on p. 77: for a map $f:V\to W$ and a tensor $T\in \mathcal{T}^k(W)$, the pullback of $T$ is
$f^*T\in\mathcal{T}^k(V)$ defined by $$(f^*T)(v_1,\ldots, v_k)=T(f(v_1),
\ldots, f(v_k)).$$
Unpacking these definitions, if $v_{1p},\ldots,v_{kp}\in\mathbb{R}^n_p$,
we have $$[(f^*\omega)(p)](v_{1p},\ldots,v_{kp})=[(f_*)^* (\omega(f(p)))](v_{1p},\ldots,v_{kp})=
[\omega(f(p))](f_*(v_{1p}),\ldots,f_*(v_{kp})),$$
as claimed by Spivak.
