In Michael Spivak's book, Calculus on Manifolds, in 4th Chapter theorem 8 (4-8), I am having a difficulty in making sense of the expression, $f^{*}\omega \wedge f^{*}\eta$.
Here $f:\mathbb{R}^{n}\to \mathbb{R}^{m}$ is a differentiable function.
$Df(p): \mathbb{R}^{n}\to \mathbb{R}^{m}$ is the derivative at $p$ and a linear transformation induced by this is $f_{*}: \mathbb{R}^{m}_{f(p)} \to \mathbb{R}^{n}_{p}$ such that $f_{*}(v_{p}) = (Df(p)(v))_{f(p)}$.
$f^{*}\omega(p)((v_{1})_{p},\ldots,(v_{k})_{p}) = w(f(p))(f_{*}(v_{1})_{p}),\ldots,f_{*}((v_{k})_{p}))$
$\omega$ and $\eta $ are $k$-forms on $\mathbb{R}^{m}$.
$\omega\wedge\eta(p) = \sum_{1\leq i_{1}<\ldots i_{k}\leq n} \sum_{1\leq j_{1}<j_{2}<\ldots < j_{l}\leq n} \omega_{i_{1}i_{2},\ldots,i_{k}}(p)\eta_{j_{1}j_{2},\ldots,j_{l}}\varphi_{i_{1}}(p)\wedge\ldots\wedge\varphi_{i_{k}}(p)\wedge\varphi_{j_{1}} (p)\wedge\ldots\wedge\varphi_{j_{l}}(p)$.
From the above two notions, I am unable to make sense of $f^{*}\omega\wedge f^{*}\eta$.
Please tell me if I need to give any more clarifications about the question.