help understanding definition of pullback and calculating it let $w = A(x,y,z)dy\wedge dz$ where $y = h(x,z)$ and $(x,z) \in D = [0,1]^2$. Let $c$ be the singular 2-cube on S, i.e. $c: [0,1]^2 \to S$ is continuous where $S \subset \mathbb{R^3}$
I am trying to calculate the pullback $c^{\star}w$
from definition of pullback I have that $c^\star(A) dy \wedge dz = A(c)(c^\star(dy \wedge dz))$ but I am unable ot proceed.. perhaps I have the wrong defintion. The answer should be $-A(x,h(x,z),z)\frac{\partial h}{\partial x}$
 A: I will assume that $S$ is a surface defined by $y=h(x,z)$ where $h$ is a smooth function, and $x,z$  is a coordinate frame on $S$. Further, $\omega$ is probably a $2$-form on $S$ defined by
$$\omega(x,z):=A(x,h(x,z),z) (h_x\,\mathrm{d}x + h_z\mathrm{d}z)\wedge dz=A(x,h(x,z),z) h_x\,\mathrm{d}x\wedge \mathrm{d}z$$
where $A(x,y,z)$ is a function $\mathbb{R}^3\to \Bbb R$. This can be rewritten to $$-A(x,h(x,z),z)\frac{\partial h}{\partial x} \mathrm{d}z\wedge \mathrm{d}x$$ which looks similar to your expression: however, this still deals only with a reformulation of $\omega$ and not yet with $c$. Note that $\omega$ itself is the pullback of the form $\tilde\omega:=A(x,y,z) \mathrm{d}y\wedge \mathrm{d}z$ as a form on $\Bbb R^3$ induced by inclusion $\iota: S\hookrightarrow \Bbb R^3$, that is, $\omega=\iota^*\tilde\omega$.
As for $c^*(\omega)$, you can assume that there are functions $c^x, c^z: [0,1]^2\to\Bbb R$ such that $c=(c^x, h(c^x,c^z), c^z)$. Then
$$
c^*\omega=-A(c^x, h(c^x, c^z), c^z) \frac{\partial (h\circ (c^x,c^z)))}{\partial x} \mathrm{d}c^z\wedge \mathrm{d}c^x
$$
which could be further expanded using chain rules etc.. But I'm not sure if this is what you want.
