In the proof of Faraday's law, the following identity is used:
$$ \frac{d}{dt} \left[ \int_{\sum(t)} B(t) \cdot dA \right]|_{t_o}= \int_{\sum(t_o)} \partial_t B|_{t_o} \cdot dA + \frac{d}{dt} \int_{\sum(t)} B(t_o) \cdot dA $$
How would I wrote the above identity in the language of differential forms? The main point I'm trouble with is how to bring in the time derivative into differential forms since we only have the exterior derivative.
