I've been doing some work with differential forms on manifolds. I am still new to the subject so please excuse me for any minor gaffes. Currently, every discussion of manifolds I have done is with them in some ambient space, usually a dim $k$ manifold in $R^n$. Now, (and I think this is why some books say it is often easier to work without an ambient space).
Now, when we write a $k$ differential form on this manifold, the presence of the ambient space complicates things. Specifically, if I have a form defined on some coordinate patch of a k surface in $R^n$, the form has to be written as: $$ \sum_I f_I dx_I $$ But, if we have a diffeomorphism, we can use this diffeomorphism to pull-back the form to a top form in $R^k$, giving it the neat representation $f dx_1 \wedge \cdots \wedge dx_k$. My question is: is this pulled back form "equal" to the form in the ambient space? It seems like there would be some differences, for example, where the forms are being evaluated (evaluation being the k vectors that the form takes in and spits out a real number with).